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Colloquium: Search for a drifting proton-electron massratio from H2

W. Ubachs and J. Bagdonaite

Department of Physics and Astronomy, LaserLaB, VU University,De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands

E. J. Salumbides

Department of Physics and Astronomy, LaserLaB, VU University,De Boelelaan 1081, 1081 HV Amsterdam, The Netherlandsand Department of Physics, University of San Carlos,Cebu City 6000, Philippines

M. T. Murphy

Centre for Astrophysics and Supercomputing, Swinburne University of Technology,Melbourne, Victoria 3122, Australia

L. Kaper

Astronomical Institute Anton Pannekoek, Universiteit van Amsterdam,Postbus 94249, 1090 GE Amsterdam, The Netherlands

(published 4 May 2016)

An overview is presented of the H2 quasar absorption method to search for a possible variation of theproton-electron mass ratio μ ¼ mp=me on a cosmological time scale. The method is based on acomparison betweenwavelengths of absorption lines in theH2 Lyman andWerner bands as observed athigh redshift with wavelengths of the same lines measured at zero redshift in the laboratory. For suchcomparison sensitivity coefficients to a relative variation of μ are calculated for all individual lines andincluded in the fitting routine deriving a value forΔμ=μ. Details of the analysis of astronomical spectra,obtained with large 8–10 m class optical telescopes, equipped with high-resolution echelle gratingbased spectrographs, are explained. The methods and results of the laboratory molecular spectroscopyof H2, in particular, the laser-based metrology studies for the determination of rest wavelengths of theLyman and Werner band absorption lines, are reviewed. Theoretical physics scenarios delivering arationale for a varying μ are discussed briefly, as well as alternative spectroscopic approaches to probevariation of μ, other than theH2method.Also a recent approach to detect a dependence of the proton-to-electron mass ratio on environmental conditions, such as the presence of strong gravitational fields, arehighlighted. Currently some 56 H2 absorption systems are known and listed. Their usefulness to detectμ variation is discussed, in terms of column densities and brightness of background quasar sources,along with future observational strategies. The astronomical observations of ten quasar systemsanalyzed so far set a constraint on a varying proton-electron mass ratio of jΔμ=μj < 5 × 10−6 (3σ),which is a null result, holding for redshifts in the range z ¼ 2.0–4.2. This corresponds to look-backtimes of ð10–12.4Þ × 109 years into cosmic history. Attempts to interpret the results from these ten H2

absorbers in terms of a spatial variation of μ are currently hampered by the small sample size and theircoincidental distribution in a relatively narrow band across the sky.

DOI: 10.1103/RevModPhys.88.021003

CONTENTS

I. Introduction 2II. Theories of a Varying μ 3III. The Spectrum of H2 as a Test Ground 4

A. Laboratory investigations of molecular hydrogen 5B. Sensitivity coefficients 6C. The H2 molecular physics database 8

IV. Astronomical Observations of H2 8A. Spectral observations and instrumental effects 9B. Fitting of spectra 10

C. Known H2 absorption systems at moderate-to-highredshift 11

D. Constraints from individual quasars 13E. Current high-redshift constraint on Δμ=μ 15F. Spatial effect on μ 15

V. Perspectives 16VI. Other Approaches to Probe μ Variation 18

A. Gamma-ray bursts 18B. Chameleon fields: Environmental dependences of μ 18C. μ variation in combination with other fundamental

constants 19

REVIEWS OF MODERN PHYSICS, VOLUME 88, APRIL–JUNE 2016

0034-6861=2016=88(2)=021003(23) 021003-1 © 2016 American Physical Society

http://dx.doi.org/10.1103/RevModPhys.88.021003




D. Radio astronomy 19VII. Conclusion 20Acknowledgments 20References 21

I. INTRODUCTION

For a long time scientific thought had been rooted in thebelief that the laws of nature are universal and eternal: they arethe rules responsible for the evolution of every part of theUniverse but they are never subject to a change themselves, nordo they depend on a specific location. Physical laws areencrypted with the so-called fundamental constants that areregarded as fundamental because they can be determined onlyby experiment. There is no theory predicting the values of thefundamental constants, while at the same time their specificvalues are finely tuned to accommodate a universe withcomplexity (Carr and Rees, 1979). Testing the immutabilityof the laws is equivalent to probing space-time variations andlocal dependences of the values of the fundamental constants.Dirac (1937) was one of the first who questioned whether theconstants are simply mathematical numbers or if they can bedecoded and understood within a context of a deeper cosmo-logical theory. As a result of this reasoning he postulated thatone of the fundamental constants, the constant describing thestrength of gravity, would vary over cosmic time. The specifichypothesis of the scaling relationG ∝ 1=t has been extensivelydiscussed and falsified over time (Barrow, 2002).Today, given the substantial body of theoretical and

experimental work produced over the past decades, theinvariance of physical laws hardly classifies as an axiomanymore, but rather is a testable and therewith falsifiablescientific question with exciting implications and vigorousresearch programs. The search for varying fundamentalconstants, or for putting constraints on their variation, hasbecome part of experimental science. Tests for variations offundamental constants are focused on dimensionless constantsfor the searches to be operationally viable (Uzan, 2011).Important milestones were achieved in the domain of obser-vational astrophysics, such as establishing the alkali-doubletmethod to probe the constancy of the fine-structure constant αusing cosmologically distant objects (Savedoff, 1956; Bahcalland Schmidt, 1967), followed by the more general and moreaccurate many-multiplet method (Dzuba, Flambaum, andWebb, 1999). The latter method is at the basis of theground-breaking studies on α variation on a cosmologicaltime scale by Webb et al. (1999), as well as the search for aspatial effect on varying constants (Webb et al., 2011).Thompson (1975) suggested that thewavelengths ofmolecu-

lar hydrogen transitions could be used to probe a possiblevariation of the ratio of the proton-electron inertial masses.Initial studies to probe H2 at high redshift employed telescopesin the 4 m class; the multiple mirror telescope (MMT) atArizona, the Anglo-Australian telescope in Australia, and theCerro Tololo Inter-American Observatory at La Serena, Chile.Thedevelopment of the 8–10mclass optical telescopes [Keck atHawaii and the European Southern Observatory very largetelescope (ESO-VLT) at Paranal, Chile] constituted an instru-mental basis required to implement the aforementioned meth-ods in extragalactic studies at high precision.

Although the analysis of big bang nuclear synthesis ofelements (Kolb, Perry, and Walker, 1986; Barrow, 1987), ofcosmic microwave background radiation patterns (Landau andScóccola, 2010), of isotopic composition of ores found in theOklo mine in Gabon (Shlyakhter, 1976), and of elementalabundances in meteorites (Olive et al., 2002) has resulted inconstraints on varying constants, spectroscopic methods cur-rently are the preferred testing grounds for probing a possiblevariation of a fundamental constant. The main reason is thatfrequency or wavelength measurements can be performed atextreme precision. This holds, in particular, for advancedlaboratory metrology studies with ultrastable lasers, ion traps,laser-cooling methods, frequency-comb lasers, and atomicfountain clocks as ingredients. A simultaneous measurementof optical transitions in Hgþ and Alþ ions has produced a limiton a drift rate for the fine-structure constant in the present epochof _α=α ¼ ð−1.6� 2.3Þ × 10−17 yr−1 (Rosenband et al., 2008).Recently, laboratory constraints on both α and μ have beendetermined bymeasuring two optical transitions in 171Ybþ ions(Godun et al., 2014;Huntemann et al., 2014).Assuming a lineardrift rate, somewhat intertwined constraints of _μ=μ<10−16yr−1

and _α=α < 10−17 yr−1 were derived. A direct μ constraint wasderived from a molecular study (measuring a transition in SF6)resulting in _μ=μ ¼ ð−3.8� 5.6Þ × 10−14 yr−1 (Shelkovnikovet al., 2008), which is less constraining but also less modeldependent (Flambaum and Tedesco, 2006).Astrophysical approaches bear the advantage that very large

time intervals are imposed on the measurements of transitionwavelengths at high redshift, leading to an enhancement insensitivity by 109–1010 for probing a rate of change, withrespect to pure laboratory searches. This is under the assumptionthat a fundamental constant would vary linearly in time. Searchfor a variation ofα is generally investigated via themeasurementof atomic lines and variation of μ through the measurement ofmolecular lines. As will be discussed, the H2 lines are not themost sensitive probes. Other molecules, in particular, theammonia (Flambaum and Kozlov, 2007) and methanol(Jansen et al., 2011) molecules, are more sensitive testinggrounds to probe a variation ofμ, butH2 has the advantage that itis ubiquitously observed in theUniverse, andup to high redshift.The B1Σþ

u − X1Σþg Lyman and C1Πu − X1Σþ

g Werner bandlines are strong dipole-allowed absorption lines of the H2

molecule covering the wavelength window λ ¼ 910–1140 Å.These wavelengths cannot be observed with ground-basedtelescopes in view of the opaqueness of the Earth’s atmospherefor wavelengths λ < 3000 Å. When observing the Lyman andWerner band lines at redshifts z, their wavelengths aremultipliedby a factor of (1þ z) due to the expansion of space on acosmological scale. Hence, for redshifts z≳ 2 the H2 lines areshifted into the atmospheric transparency window so that thelines can be observed from ground-based telescopes, such as theVLTof the ESO or the Keck telescope onMauna Kea at Hawaii.Alternatively, the H2 absorptions can be observed from outsidethe Earth’s atmospherewith the Hubble Space Telescope (HST).In the latter caseH2 absorption at redshifts z < 2 can bedetected.The main focus of this Colloquium is on a variation of the

proton-electron mass ratio μ, a dimensionless constant whosevalue from laboratory experiments is known with a 4 × 10−10

relative precision and listed in the CODATA 2010 release of

Ubachs et al.: Colloquium: Search for a drifting proton- …

Rev. Mod. Phys., Vol. 88, No. 2, April–June 2016 021003-2

the recommended values of the fundamental physical con-stants (Mohr, Taylor, and Newell, 2012). A fourfold improve-ment over the precision of the current recommended value of μwas reported recently (Sturm et al., 2014), leading to

μ≡mp

me¼ 1836.152 673 77 ð17Þ: ð1Þ

It should be realized that searches for a varying constant μ donot necessarily imply a determination of its value; in fact, innone of the studies performed so far was this the strategy.Rather, a variation of μ is probed as a differential effect

Δμ ¼ μz − μ0; ð2Þ

where μz is the value of the proton-electron mass ratio at acertain redshift z and μ0 is the value in the current epoch, atzero redshift. The differential effect is investigated only inrelative terms, i.e., values of Δμ=μ are determined in thestudies (Jansen, Bethlem, and Ubachs, 2014).In this Colloquium an account is given of the high-redshift

observations of H2 absorption spectra in the sight line ofquasar1 sources and the constraints on a varying proton-electron mass ratio that can be derived from these observationswhen comparing to laboratory spectra of H2. Also a review isgiven of the laser-spectroscopic investigations of the Lymanand Werner band lines as performed in the laboratory, for boththe H2 and hydrogen deuteride (HD) isotopomers. Limitationsand prospects of the astronomical observations and the H2

method will be discussed, while a status report is presented onthe objects suitable for such studies.

II. THEORIES OF A VARYING μ

A variety of theoretical scenarios exists in which funda-mental constants are allowed to vary: unification theories thatinclude extra dimensions and theories embracing fundamentalscalar fields. Theories postulating additional space-timedimensions date back to the Kaluza-Klein concept of unifyingfield theories of electromagnetism and gravity in five dimen-sions. Modern string theories are formulated in as many as tendimensions and postulate a Klein compactification (Klein,1926) to comply with the perceivable four-dimensionaluniverse. In this dimensional reduction, the conventionalconstants appear as projected effective parameters that maybe easily varied within the context of cosmological evolutionof the Universe, resulting in variations of the effectiveconstants as measured in four-dimensional space-time(Martins et al., 2010). Interestingly, the observation of varyingconstants has important implications in the present debate onstring theory landscapes or multiverses (Schellekens, 2013).In another class of theories, introduced by Bekenstein

(1982), a cosmological variation in the electric charge e isproduced by an additional scalar field, or a dilaton field, whichmay weakly couple to matter. A more general approach wasproposed by Barrow, Sandvik, and Magueijo (2002) (Sandvik,

Barrow, and Magueijo, 2002), who created a self-consistentcosmological model with varying α. In this model anysubstantial α changes are suppressed with the onset of darkenergy domination, a concept with far-reaching implicationsfor laboratory searches of varying constants. Barrow andMagueijo (2005) subsequently described a scenario that, byinducing variations in the mass parameter for the electron,specifically addresses variation of μ. In these theories, avariation of constants is driven by the varying matter densityin the Universe. This led to the development of the so-calledchameleon scenarios where inhomogeneities in the constantscould be observed by probing environments with very lowlocal mass densities, in contrast to the high-density environ-ment of Earth-based experiments (Khoury and Weltman,2004). Such scenarios of environmental dependences offundamental constants were extended to cases where α andμ were predicted to depend on local gravitational fields(Flambaum and Shuryak, 2008), by definition implying abreakdown of the equivalence principle.As an alternative to scenarios producing a variation of

constants due to massless dark-energy-type fields, Stadnik andFlambaum (2015) recently linked the variation to a condensateof massive dark-matter particles. It is stipulated that thiscauses a slow drift as well as an oscillatory evolution of theconstants of nature.In the above-mentioned theories by Barrow and co-workers

(Barrow, Sandvik, and Magueijo, 2002; Sandvik, Barrow, andMagueijo, 2002) it is implied that the values of fundamentalconstants are connected to the matter density Ωm and darkenergy density ΩΛ composition of the Universe. The H2

absorption method is typically applied for redshifts z > 2, i.e.,for a universe with a matter density of Ωm > 0.9 (includingdark matter). In contrast, the laboratory searches for varying μ,such as those of Godun et al. (2014) and Huntemann et al.(2014), are performed in the present cosmic epoch withΩm ¼ 0.32, hence quite different. Even if laboratory studiesbased on atomic clocks achieve competitive constraints on arate of change of a varying constant, such as _μ=μ, this does notimply that the same physics is probed: the cosmologicalconditions setting the values of the fundamental constants arelikely to be very different.As pointed out originally by Born (1935) the fine-structure

constant α and the proton-electron mass ratio μ are the twodimensionless parameters that describe the gross structure ofatomic and molecular systems. While the fine-structureconstant α is a measure of a fundamental coupling strength,the dimensionless constant μ might be considered lessfundamental because the mass of a composite particle isinvolved. However, since the gluon field that binds quarksinside the proton is responsible for virtually all of its mass, μ issensitive to the ratio of the chromodynamic scale with respectto the electroweak scale (Flambaum et al., 2004). Thus μ maybe viewed as a parameter connecting fundamental couplingstrengths of different forces. Various theoretical scenarioshave been described relating the possible variation of bothconstants and in most schemes relying either on grandunification (Calmet and Fritzsch, 2002; Langacker, Segré,and Strassler, 2002) or on string theory (Dent and Fairbairn,2003), the rate of change in the proton-to-electron mass ratioΔμ=μ is found much larger than the rate of change in the fine

1We use the term “quasar” in this work interchangeably with theterm quasistellar object, or QSO.



structure constantΔα=α. This makes the search for a varying μa sensitive testing ground to probe for variations of funda-mental constants.In the context of spectroscopy and spectral lines the proton

and electron masses and their ratio should be considered asinertial masses, which is how masses enter the Schrödingerequation in the calculation of atomic and molecular levelstructure. The difference between inertial versus gravitationalmasses is of relevance when searching for a variation offundamental constants, since that implies a space-timedependence of the laws of physics, i.e., a violation of theEinstein equivalence principle, the fundamental concept bywhich inertial and gravitational masses are consideredidentical.

III. THE SPECTRUM OF H2 AS A TEST GROUND

As pointed out by Thompson (1975) the value of μinfluences the pattern of the rovibronic transitions in molecu-lar hydrogen (H2). A small and relative variation of the proton-to-electron mass ratio

Δμμ

¼ μz − μ0μ0

ð3Þ

will then give rise to differential shifts of the individualabsorption lines, as displayed in graphical form and vastlyexaggerated in Fig. 1. Here μz refers to the proton-electronmass ratio in a distant extragalactic system at redshift z and μ0is a reference laboratory value, i.e., a measurement at zeroredshift. A positive Δμ=μ indicates a larger μ in the distantsystem as compared to the laboratory value. Figure 1 showsthat some lines act as anchor lines, not being sensitive to avariation of μ. Most lines in the spectrum of H2 act asredshifters, thus producing a longer wavelength for a highervalue of μ. Only a minor fraction of lines acts as blueshifters,

such as the lines indicated with L0P3 andW0P3, correspond-ing to the Pð3Þ line of the B1Σþ

u − X1Σþg (0,0) Lyman band,

and the Pð3Þ line of the C1Πu − X1Σþg (0,0) Werner band.

Here a four-digit shorthand writing is used for the H2

transitions observed in cold clouds in the line-of-sight towardquasars, such as LXPY and WXRY, where L and W refer tothe Lyman and Werner bands, X denotes the vibrationalquantum number of the excited state, P and R (or Q) referto the rotational transition, and Y indicates the ground staterotational level probed.Besides the very small differential shifts caused by a

possible variation of μ, the H2 spectral lines observed fromdistant galaxies undergo a strong redshift due to the cosmo-logical expansion of the Universe. Crucially, general relativitypredicts that all wavelengths undergo a similar redshift, henceλz ¼ ð1þ zÞλ0 throughout the spectrum. This assumption thatredshift is dispersionless underlies all analyses of varyingconstants in the early Universe based on spectroscopicmethods.When combining the effects of cosmological redshift and a

small additional effect due to a variation of the proton-electronmass ratio, the wavelength of the ith transition observed in anabsorbing system at redshift zabs is then defined as

λzi ¼ λ0i ð1þ zabsÞ�1þ Ki

Δμμ

�; ð4Þ

where λ0 is the corresponding rest wavelength, and Ki is asensitivity coefficient, discussed in Sec. III.B.In the comparison between H2 absorption spectra recorded

at high redshift and laboratory spectra, as much as possibleknown molecular physics is implemented when deriving aputative effect on Δμ=μ. The H2 molecular lines are deter-mined by a number of parameters, for each transition: (i) therest-frame wavelength λ0i which is determined in high-precision laser-based laboratory studies (see Sec. III.A); (ii) thesensitivity coefficient Ki (see Sec. III.B); (iii) the intensity,rotational line strength or oscillator strength fi; and (iv) thedamping factor or natural line broadening factor Γi (seeSec. III.C). In the analysis of astronomical spectra thismolecular physics information is combined with environmen-tal information to determine the effective line positions, linestrengths, and linewidths: (v) the redshift parameter z, (vi) theDoppler broadening coefficient b, and (vii) the columndensities NJ, which are identical for each subclass of rota-tional states J populated in the extragalactic interstellarmedium. These seven parameters determine a fingerprintspectrum of the H2 molecule. While the Doppler broadeningcoefficient bmay in many cases be assumed to be the same forall rotational levels, as is expected to result from homogeneoustemperature distributions, in some cases J-dependent valuesfor b were found (Noterdaeme, Ledoux et al., 2007; Rahmaniet al., 2013). Malec et al. (2010) discussed the fact that thisphenomenon can also be ascribed to underlying and unre-solved velocity structure on the H2 absorptions.The partition function of H2 is such that, at room temper-

ature or lower temperatures as in most extragalactic sourcesobserved (50–100 K), only the lowest vibrational quantumstate in the electronic ground state is populated: X1Σþ

g , v ¼ 0.

rest

wav

elen

gth

()

900

950

1000

1050

1100L0 P3

L1 R0

L8 R0

L6 R3

L3 R2

L10 P1

W5 Q2

W2 Q1

W0 P3

0.0 0.1 0.2 0.3 0.4 0.5

FIG. 1. Shifts in the rest wavelengths of Lyman (L) andWerner (W) lines as functions of the (exaggerated) relativevariation in μ. The solid (red) curves indicate transitions thatare redshifted with increasing Δμ=μ, while the dashed (blue)curves are blueshifters. The L1R0 transition, indicated by a dot-dashed (black) curve, represents a line with very low sensitivity toμ variation, i.e., it acts as an anchor line.



Under these conditions the lowest six rotational states J ¼0–5 are probed in the absorption spectra. Vibrational excita-tion does not play a role at these temperatures. For apopulation of only the lower quantum states the onset ofH2 absorption is at rest-frame wavelengths of λ0 ¼ 1140 Å,where the B − Xð0; 0Þ band is probed. The absorptionspectrum of H2 extends in principle to the ionization limitand beyond (Chupka and Berkowitz, 1969), so to wavelengthsas short as 700 Å, but is truncated at 912 Å because of theLyman cutoff, i.e., the onset of the absorption continuum dueto H I.The other constraint on the coverage of the H2 absorption

spectrum is related to the atmospheric window and thereflectivity of mirror coatings generally used at large tele-scopes. The former cutoffs fall at λc ¼ 3050 Å, which meansthat an H2 spectrum is fully covered until the Lyman cutoff ifthe redshift of the absorbing galaxy is z > 2.3. For absorbingsystems at redshifts z < 2.3 the shortest wavelength lines liebeyond the onset of atmospheric opacity and will notcontribute to the observable spectrum. All this means thatin the best case the Lyman bands of H2 can be followed up toB − Xð18; 0Þ and the Werner bands up to C − Xð4; 0Þ.

A. Laboratory investigations of molecular hydrogen

The strongest absorption systems of molecular hydrogencorrespond to the electronic excitation of one of the 1s groundstate orbitals to an excited 2pσ orbital or to the twofolddegenerate 2pπ orbital. In the former case the B1Σþ

u molecularsymmetry state is probed via the Lyman system(B1Σþ

u − X1Σþg ). In the latter case the C1Πu molecular sym-

metry state is excited in the Werner system (C1Πu − X1Σþg ).

For each vibrational band, the rotational series or branchesare labeled according to the change in rotational angularmomentum ΔJ ¼ J0 − J00, where J0 and J00 are the rotationalquantum numbers of the upper and lower levels, respectively.Thus the PðJ00Þ branch is for transitions with ΔJ ¼ −1, QðJ00Þis for ΔJ ¼ 0, and RðJ00Þ for ΔJ ¼ þ1. For reasons ofmolecular symmetry in the Lyman bands only P and R linesoccur, while all three types occur in the Werner bands.There is an extensive data set of emission measurements of

the Lyman (Abgrall et al., 1993a) and Werner bands (Abgrallet al., 1993b) that were analyzed with a classical 10 mspectrograph. Upon averaging over all emission bands thisresulted in values for level energies B1Σþ

u ; v; J and C1Πu; v; Jat an absolute accuracy of 0.15 cm−1 (Abgrall et al., 1993c),which corresponds to Δλ=λ ≈ 1.5 × 10−6 at λ ¼ 1000 Å.From these data a comprehensive list of absorption lines inthe B − Xðv0; 0Þ Lyman and C − Xðv0; 0Þ Werner bands, alloriginating from the ground vibrational level, is calculated.This forms a backup list at an accuracy of 0.15 cm−1 to beused for those lines where no improved laser-based calibra-tions have become available.More recently laser-based laboratory studies of molecular

hydrogen were carried out employing direct extreme ultra-violet (XUV) laser excitation (see left panel of Fig. 2) ofrotational lines in the Lyman and Werner bands of the H2

molecule. The measurements were performed using a narrow-band and tunable laser system in the visible wavelength range,

which is upconverted via frequency doubling in crystals andthird-harmonic generation in xenon gas jets delivering coher-ent radiation in the range 900–1150 Å. The method of 1þ 1

resonance-enhanced multiphoton ionization was employed fordetection of the spectral resonances. The spectroscopicresolution was as accurate as 0.02 cm−1, achieved via sub-Doppler spectroscopy in a molecular beam. The absolutecalibration of the H2 resonances was performed by interpo-lation of the frequency scale in the visible range by a stabilizedetalon and by comparing to a saturated absorption spectrum ofmolecular iodine (Velchev et al., 1998; Xu et al., 2000). Anexample of a recording of the L15R2 line is displayedin Fig. 3.More than 160 H2 spectral lines in the B − Xðv0; 0Þ Lyman

(v0 up to 19) and C − Xðv0; 0Þ Werner bands (v0 up to 4) werecalibrated. Only the vibrational ground state v ¼ 0was probedand in most cases J ¼ 0–5 rotational states, although in somecases only the very lowest rotational quantum states. Theresults, obtained at a typical accuracy of Δλ=λ ¼ 5 × 10−8,were published in a sequence of papers covering the relevantwavelength range 900–1150 Å (Philip et al., 2004; Ubachsand Reinhold, 2004; Hollenstein et al., 2006; Reinhold et al.,2006; Ivanov, Vieitez et al., 2008). In cases where only thepopulation of the lowest rotational quantum states J ¼ 0–2could be probed, mainly for the R-branch lines, the wave-length positions of PðJÞ lines were calculated to high accuracyfrom rotational combination differences in the X1Σþ

g , v ¼ 0

ground state, derived from the precisely measured pure

FIG. 2. Excitation schemes used to determine the Lyman andWerner transition wavelengths. Left panel: A direct excitation(a) scheme from X to B, C states using a narrowband XUV-lasersystem (or, alternatively, detection of XUVemission from excitedB and C states). Right panel: An indirect determination scheme isrepresented comprising two-photon Doppler-free excitation (b) inthe EF − X system and Fourier-transform emission measure-ments (c) of the EF − B and EF − C systems.



rotational spectrum of H2 in the far infrared (Jennings, Bragg,and Brault, 1984).Similar XUV-laser studies were performed for the HD

molecule (Ivanov, Roudjane et al., 2008), achieving anaccuracy of Δλ=λ ¼ 5 × 10−8, while a more comprehensivestudy on HD was also performed using vacuum ultraviolet(VUV)-Fourier-transform absorption spectroscopy withsynchrotron radiation yielding a lower accuracy of Δλ=λ ¼4 × 10−7 (Ivanov et al., 2010).An improved laboratory wavelength determination of the

Lyman and Werner bands is derived from a combination oflaser-based deep-UV (DUV) laser spectroscopy and Fourier-transform (FT) spectroscopy (Salumbides et al., 2008; Baillyet al., 2010). The advantage of this two-step method,schematically depicted in the right-hand panel of Fig. 2, isthat it bypasses the instrument bandwidth limitation in thedirect XUV-laser spectroscopy described earlier. In the FTstudies, emission from a low-pressure H2 microwave dis-charge was sent into an FT spectrometer, and a set of filtersand detectors were used to cover the broad wavelength rangebetween 5 μm and 4500 Å (2000–22000 cm−1). A small partof the obtained FT spectrum is plotted in Fig. 4(b). Thecomprehensive FT spectra include transitions belonging todifferent vibrational bands of (symmetry-allowed) pair com-binations of electronic states: B1Σþ

u , C1Πu, B01Σþu , EF1Σþ

g ,GK1Σþ

g , H1Σþg , I1Πg, and J1Δg. The accuracy of the line

position determination varies for the different wavelengthranges, ultimately limited by Doppler broadening. For stronglines in the infrared range accuracies of 0.0001 cm−1 could beobtained.To connect the level structure of the excited electronic state

manifold to the X1Σþg v ¼ 0, J ¼ 0 level, results from

Doppler-free two-photon spectroscopy on the EF − X (0,0)band are used (Hannemann et al., 2006; Salumbides et al.,2008). In these laser spectroscopic investigations, a pulsednarrowband laser source, frequency-comb assisted frequency

calibration, interferometric alignment for Doppler-shift sup-pression, frequency-chirp measurement, and ac-Stark correc-tion are all essential ingredients to achieve the ∼10−9accuracies (Hannemann et al., 2006). The Qð0Þ and Qð1Þlines, with accuracies better than 2 × 10−4 cm−1, are specifi-cally used as anchor lines for para- and ortho-H2, respectively.A recording of the Qð0Þ line is plotted in Fig. 4(a).A comparison of the QðJ00 ¼ 2 − 5Þ transitions measuredin the DUV investigations with those derived from theFT investigations confirmed the accuracy of the two-stepmethod of obtaining the Lyman (Δλ=λ ∼ 10−9) and Werner(Δλ=λ ∼ 10−8) transition wavelengths.

B. Sensitivity coefficients

In comparing astrophysical spectra, yielding the wave-lengths at high redshift λzi , with laboratory spectra providingλ0i , a fit can be made to extract or constrain a value ofΔμ=μ viaEq. (4). HereKi is the sensitivity coefficient, different for eachtransition, and defined by

Ki ¼d ln λid ln μ

. ð5Þ

This equation can also be expressed in terms of the quantumlevel energies Eg and Ee of ground and excited states involvedin a transition and a spectroscopic line:

Ki ¼ −μ

Ee − Eg

�dEe

dμ−dEg

dμ

�: ð6Þ

Note that some calculate sensitivity coefficients to a possiblevariation of the electron-proton mass ratio μ0 ¼ me=mp ¼1=μ, in which case the values of the Ki reverse sign (Kozlovand Levshakov, 2013). Also sometimes the sensitivity coef-ficients are defined in connection to frequencies (Jansen,Bethlem, and Ubachs, 2014), Δν=ν ¼ KΔμ=μ, in which casethe values of the Ki also reverse sign.

(a)

Q0EF(0)-X(0)

E + 99164 (cm-1)0.784 0.786 0.788 0.79

(b)

EF(9)-B(2)

e(3)-a(2)

I-(1)-C-(1)

GK(5)-C-(2)

R4

R0

P3

P2 P2

GK(3)-C+(1)R2

EF(6)-B(1)

E + 12490 (cm-1)5 10 15 20 25 30

FIG. 4. (a) Laser spectroscopic recording of the Qð0Þ two-photon transition in theEF1Σþ

g − X1Σþg (0,0) band. (b) Recording

of part of the Fourier-transform spectrum of the EF1Σþg − B1Σþ

u

emission.

fundamental (cm-1)17718.50 17718.55 17718.60 17718.65 17718.70 17718.75

*etalon

I2

H2

XUV wavenumber (cm-1)106311.0 106311.3 106311.6 106311.9 106312.2 106312.5

FIG. 3. Recording of the XUV-absorption spectrum of L15R2,i.e., the Rð2Þ line in the B1Σþ

u − X1Σþg (15,0) Lyman band (lower

spectrum) with the I2 reference spectrum (middle spectrum) andetalon markers (top spectrum) for the calibration. The line markedwith an asterisk is the a1 hyperfine component of the Rð66Þ linein the B − Xð21; 1Þ band in I2 at 17 718.7123 cm−1. FromUbachs et al., 2007.



The sensitivity coefficient Ki can be considered as anisotope shift of a transition in differential form. Its value can ingood approximation be understood in the Born-Oppenheimerapproximation, separating the contributions to the energy ofthe molecule

Ei ¼ Eelec þ Evibrðμ−1=2Þ þ Erotðμ−1Þ; ð7Þ

with Eelec the electronic, Evibr the vibrational, and Erot therotational energy. The dependence of these energy terms on μis then known: (i) in the Born-Oppenheimer approximationthe electronic energy is independent of μ, (ii) in a harmonicoscillator approximation the vibrational energy scales asð1= ffiffiffi

μp Þ, and (iii) in a rigid rotor approximation the rotational

energy scales as 1=μ. These scalings explain the values of thesensitivity coefficients. For the (0,0) bands in the LymanB − X and Werner C − X systems the transition is almost fullyelectronic in nature and henceforth Ki ≈ 0. For ðv0; 0Þ bandsprobing higher vibrational energies a fraction of the excitedstate energy becomes vibrational in nature and therefore thevalue of Ki increases. Because the amount of vibrationalenergy remains below 15% the value of Ki will remain below0.07, where it is noted that for pure vibrational energyKi ¼ −1=2. Similarly for the fraction of rotational energyKi ¼ −1 (Jansen, Bethlem, and Ubachs, 2014).The K values can be derived in a semiempirical approach,

separating electronic energies and expressing rovibrationalenergies in a Dunham representation:

Eðv; JÞ ¼Xk;l

Yk;l

�vþ 1

2

�k½JðJ þ 1Þ − Λ2�l ð8Þ

with Dunham coefficients Yk;l, v and J vibrational androtational quantum numbers, and Λ signifying the electronicorbital momentum of the electronic state (0 for the X1Σþ

g andB1Σþ

u states and 1 for the C1Πu state). This method drawsfrom the advantage that the energy derivatives dE=dμ inEq. (6) can be replaced by derivatives dY=dμ and that thefunctional dependence of the Dunham coefficients Yk;l isknown (Reinhold et al., 2006; Ubachs et al., 2007).Varshalovich and Levshakov (1993) were the first to calculateK coefficients based on such a Dunham representation of levelenergies of ground and excited states. Thereafter Ubachs et al.(2007) recalculated K coefficients from improved accuracylaser-based measurements of Lyman and Werner bands withinthe Dunham framework. They also showed how to correct theK values for adiabatic and nonadiabatic effects in the excitedstates.Alternatively, numerical values for the sensitivity coeffi-

cients can be derived by making use of ab initio calculationsfor the hydrogen molecule as pursued by Meshkov et al.(2006), yielding values KAI . These are found to be in goodagreement with the values KSE obtained in the semiempiricalapproach (Ubachs et al., 2007), within an accuracyof ΔK ¼ jKAI − KSEj < 3 × 10−4.Recently, improvements were made following both the

semiempirical and the ab initio approaches (Bagdonaite,Salumbides et al., 2014). For the semiempirical analysis itwas realized that fitting the Dunham coefficients is not

necessary. Instead, derivatives of the level energies dE=dμcan be obtained from numerical partial differentiation withrespect to the vibrational v and rotational J quantum numbers

dEdμ

��v;J

¼ −1

2μ

�vþ 1

2

� ∂E∂v

��v;J

−1

μ

JðJ þ 1Þ2J þ 1

∂E∂J

��v;J: ð9Þ

This provides a more direct procedure only requiring deriv-atives to calculate sensitivity coefficients Ki, while in practicethe calculation of derivatives appears to be more robust thancalculating strongly correlated Dunham coefficients.Further, an improved round of ab initio calculations were

carried out including the best updated numerical representa-tions of the four interacting excited state potentials for B1Σþ

u ,B01Σþ

u , C1Πu, and D1Πu states (Staszewska and Wolniewicz,2002; Wolniewicz and Staszewska, 2003), including adiabaticcorrections and the mutual nonadiabatic interactions(Wolniewicz, Orlikowski, and Staszewska, 2006). Thesecalculations were performed for a center value for the reducedmass of μred ¼ 0.5μ for H2 and ten different intermediatevalues following an incremental step size for Δμ=μ of 10−4

around the center value. Taking a derivative along the μ scaleyields level sensitivity coefficients μdE=dμ for the excitedstates. A similar procedure was followed for the X1Σþ

g groundstate based on the Born-Oppenheimer potential computed byPachucki (2010) and nonadiabatic contributions of Komasaet al. (2011). Accurate values for sensitivity coefficients werederived by dividing through the transition energies, as in thedenominator of Eq. (6).Values for the Ki sensitivity coefficients pertaining to

transitions originating in the lowest rotational quantum statesare plotted in Fig. 5 for a selected number of Lyman andWerner bands. These values obtained from recent ab initiocalculations are in very good agreement with the updatedsemiempirical calculation employing Eq. (9). The negativevalues forW0 and L0 bands, due to the fact that the zero-pointvibration in the ground state is larger than in the excited states,makes those lines blueshifters: they shift to shorter wave-lengths for increasing μ. The irregularities in the data pro-gressions, such as for the Rð5Þ line in the W0 band, show thelocal mutual (nonadiabatic) interactions between B1Σþ

u andC1Πu states, giving rise to state mixing. The lines in the L19band exhibit the largest K coefficients, because the fraction ofvibrational energy to the total energy in the excited state islargest. Higher vibrational bands than L19 in the Lymansystem are not considered because they are at a rest wave-length λ < 910 Å and therewith beyond the Lyman cutoff ofatomic hydrogen. Sensitivity coefficients for the HD moleculewere also calculated through the ab initio approach (Ivanovet al., 2010).In Fig. 6 the sensitivity coefficients to the μ variation are

plotted as a function of wavelength. The size of the data pointsreflects the line oscillator strength fi for the H2 lines, hencerepresenting their significance in a μ-variation analysis. Thesignificance of the various spectral lines in a μ-variationanalysis depends further on the population distribution in theabsorbing cloud as well as on the total column density. Atypical population temperature is ≈50–100 K, and thereforethe population of H2 is dominantly in the J ¼ 0 and J ¼ 1



para and ortho ground states. However, the population ofrotational quantum states of H2 in the absorbing gas is likely todepart from thermodynamic equilibrium, so that the popula-tion distribution reflects some combination of the prevailingkinetic temperature, the radiation environment in the absorb-ing region, and the rates of chemical processes involving H2.

This is contrary to the case of CO which is found to reflect apartition function commensurate with the local cosmic back-ground temperature TCO ¼ ð1þ zÞTCMB (Noterdaeme et al.,2011). HD is so far detected only in the J ¼ 0 state (Ivanchiket al., 2010; Noterdaeme et al., 2010; Tumlinson et al.,2010). In case of very large column densities, i.e., forlogN½H2=cm−2� > 18, the low-J levels will be saturatedand broadened, making them less suitable to determine linecenters. At these high column densities the spectral lines withJ ≥ 1 and even with J ≥ 3 will most decisively contribute toconstraining Δμ=μ.The distribution of points in Fig. 6 demonstrates that

K coefficients strongly correlate with wavelength for eachband system separately. Observation of both Werner andLyman systems counteracts the wavelength correlation ofthe K values and makes the analysis more robust.

C. The H2 molecular physics database

The extraction of information from high-redshift observa-tions of H2 molecular spectra depends on the availability of adatabase including the most accurate molecular physics inputfor the relevant transitions in the Lyman and Werner absorp-tion bands. Values for the wavelengths λi were collected fromthe classical data (Abgrall et al., 1993a, 1993b), the directXUV-laser excitation (Philip et al., 2004; Hollenstein et al.,2006; Reinhold et al., 2006; Ivanov, Vieitez et al., 2008), andthe two-step excitation process (Salumbides et al., 2008;Bailly et al., 2010); in each case the most accurate wavelengthentry is adopted. The Ki sensitivity coefficients are includedfrom the recent ab initio calculations involving a four-stateinteraction matrix for the excited states (Bagdonaite,Salumbides et al., 2014). In order to simulate the spectrumof molecular hydrogen values for the line oscillator strength fiand radiative damping parameter Γi, giving rise to Lorentzianwings to the line shape, should also be included for all H2 linesin the Lyman and Werner bands. These were calculated byAbgrall, Roueff, and Drira (2000). The recommended data-base containing the molecular physics parameters for H2 andHD needed to model quasar absorption spectra was publishedas a supplementary file by Malec et al. (2010).

IV. ASTRONOMICAL OBSERVATIONS OF H2

While the earliest observations of H2 at high redshift alongquasar sight lines were performed in the 1970s (Aaronson,Black, and McKee, 1974; Carlson, 1974), the first study ofμ variation using the H2 method was carried out somewhatlater by Foltz, Chaffee, and Black (1988) using the MMT(Arizona) composed of six dishes of 1.8 m. When the 8–10 mclass telescopes became available, such as the Keck telescopein Hawaii and the ESO VLT at Paranal (Chile) both equippedwith high-resolution spectrographs, systematic investigationsof H2 absorption systems at intermediate and high redshiftswere pursued.An example quasar spectrum of the source B0642 − 5038

recorded with the ultraviolet and visible echelle spectrograph(UVES) mounted on the VLT (Noterdaeme, Ledoux et al.,2008) (raw data available in the ESO archive) is displayed inFig. 7. A characteristic feature of such a spectrum is the strong

LymanWerner

Ki

-0.02

0.00

0.02

0.04

rest wavelength ( )950.0 1000.0 1050.0 1100.0

FIG. 6. Sensitivity coefficients Ki for μ variation in H2 as afunction of rest-frame wavelengths. Solid (blue) symbols areassociated with Lyman lines and open symbols to Werner lines.The size of the data points reflects line oscillator strengths fi forthe spectral lines.

L8

W2

L15

L19

W5

L10

L5

L0

W0

Ly RLy PWer QWer PWer R

Ki

-0.020

0.000

0.020

0.040

0.060

J''0 1 2 3 4 5 6

FIG. 5. Sensitivity coefficients Ki of individual lines in theLyman and Werner bands showing the range of values fromaround −0.02 to 0.06. Solid (blue) symbols are associated withLyman lines and open (red) symbols to Werner lines. From L8,level crossings occur between levels in the B1Σþ and C1Πþstates, leading to irregularities in the K progression of the LymanP, R and Werner Q transitions.



and broad central H I Lyman-α emission peak at 4980 Å,yielding a value for the redshift of the quasar (z ¼ 3.09). Alsoa weaker emission peak related to Lyman β is found near4250 Å, as well as a C IVemission peak at 6300 Å. The broadabsorption feature exhibiting Lorentzian wings at 4450 Å isthe “damped Lyman-α system” (DLA) found at redshiftz ¼ 2.66. DLAs are thought to mainly arise in distant, gas-rich galaxies, on the verge of producing bursts of starformation (Wolfe, Gawiser, and Prochaska, 2005). The red-shift of the DLA also determines the wavelength of the13.6 eV “Lyman-limit” cutoff in the spectrum, in this caseat 3340 Å. Shortward of this wavelength all radiation isabsorbed in the DLA. The “Lyman-α forest”—the collectionof narrower absorption lines covering the entire regionbetween the Lyman cutoff and the Lyman-α emissionpeak—arises mainly in the lower-density, filamentary struc-tures of the intergalactic medium traced by diffuse neutralhydrogen (Meiksin, 2009). The forest absorption lines are(close to) randomly distributed in redshift, giving their forest-like appearance in the spectrum. The region bluewards of theLyman-β emission peak also includes the Lyman-β absorptionlines corresponding to the forest lines at higher redshift, and soon for successive emission peaks of the hydrogen Lymanseries. Of importance to the search of a varying μ is thelocation of the H2 absorption lines. These molecular tran-sitions are related to the denser DLA cloud at redshiftz ¼ 2.66, and hence the wavelength region of Lyman andWerner H2 absorption is ð1þ zDLAÞ × ½910–1140� Å or[3340–4170] Å in the example shown in Fig. 7.

A. Spectral observations and instrumental effects

The visible (i.e., redshifted UV) observations of H2 towardbackground quasars that have so far been used to constrain

variations in μ have all been obtained using slit-fed, “echelle”spectrographs, the UVES at the VLT (Dekker et al., 2000) andthe High Resolution Echelle Spectrometer (HIRES) spec-trometer at Keck (Vogt et al., 1994). After passing through asimple slit, the quasar light is collimated onto the maindispersive element, the echelle grating, at grazing incidence.Most of the diffracted light is in high diffraction orders(∼100), each with a small spectral range (∼60 Å) but highresolving power of typically R≡ λ=FWHM ∼ 50 000 (forFWHM, the full-width-at-half-maximum resolution). To rec-ord a wide total spectral range in a single exposure, thediffracted echelle orders are “cross dispersed,” perpendicularto the echelle dispersion direction, so that many orders can besimultaneously imaged onto a rectangular charge-coupleddevice (CCD) whose width exceeds the spatial extent of theirspectral range.Observations at the Keck telescope are normally undertaken

by the investigators of the proposed observations themselves.This “visitor mode’” is sometimes also used at the VLT.However, most quasar observations at the VLT are taken in“service mode”: a local astronomer conducts the observationsand calibrations, and the data are delivered to an archive wherethe original investigators or, increasingly, other researcherscan reduce it from raw to usable form with software suppliedby the observatory or used-made improved versions of it(Murphy, Tzanavaris et al., 2007; Thompson et al., 2009).Many of the quasar observations used for μ-variation studies sofar were taken at the VLT in service mode for other purposes.Depending on the quasar brightness, observations typically

involve ∼10 separate exposures, of 1–2 hr duration each,spread over several observing nights under varying temper-ature, pressure, sky conditions, and telescope positions.A variety of calibration exposures are therefore taken atregular intervals to allow these effects and the spectrograph’s

FIG. 7. Typical spectrum of a quasar displaying the observed flux (in arbitrary units), in this case for B0642 − 5038 emitting at redshiftz ¼ 3.09 and registered by the ultraviolet and visual echelle spectrograph (UVES) mounted at the ESO VLT. The quasar spectrumdisplays the characteristic broad emission line profiles produced by, e.g., Lyman α of H I, and C IV. The Lyman-α forest arises as the lightfrom the quasar crosses multiple neutral hydrogen clouds lying in the line of sight. On the left (blue) side of the emission peak thedamped Lyman-α system at zDLA ¼ 2.66 causes a series of very strong absorption lines starting at λ ¼ ðzDLA þ 1Þ × 1216 Å ¼ 4450 Åand absorbs all the flux at λ < ðzDLA þ 1Þ × 912 Å ¼ 3340 Å producing a so-called Lyman break or Lyman-limit cutoff. Since the restwavelengths of H2 are in the UV (< 1140 Å) they are found in the Lyman-α forest. On the right (red) side of the Lyman-α emission peakthe sharp absorption lines are due to metallic ions (C IV, Fe II, Si IV, etc.) at various redshifts including that of the DLA. The absorptionsat 6800 and 7600 Å are related to the Fraunhofer B and A bands (molecular oxygen) absorbing in the Earth’s atmosphere.



configuration to be removed from the recorded spectra. Forvarying-μ studies, the most important aspect to calibrateaccurately is the wavelength scale of each quasar exposure.The standard wavelength calibration is done via comparison

with an exposure of a thorium-argon (ThAr) hollow-cathodeemission-line lamp taken in the same spectrograph configu-ration. The ThAr lamp is mounted within the spectrograph andits light illuminates the slit via a fold mirror moved intoposition to mimic the light path of the quasar photons throughthe slit. The ThAr line wavelengths are known from laboratorystudies (Lovis and Pepe, 2007), so a low-order polynomialrelationship between wavelength and pixel position on theCCD can be established by fitting the observed lines (Murphy,Tzanavaris et al., 2007). This relationship is simply assumedto apply to the corresponding quasar exposure. However, twomain differences in the light path are cause for concern(Murphy et al., 2001; Murphy, Webb, and Flambaum,2003): the ThAr light may not be aligned with the quasarlight closely enough, and the ThAr light fully illuminates theslit whereas the quasar is a point source whose image isblurred to a Gaussian-like spatial distribution by atmospheric“seeing” (refraction and turbulence effects). These effects,particularly the former, may be expected to distort the ThArwavelength scale away from the true scale for exposures ofastronomical objects over long wavelength ranges, so they areimportant to identify and correct if possible.Given that the Keck and VLT spectrographs are not actively

stabilized, it is therefore important that the ThAr exposureimmediately follows the quasar exposure, with no interveningchanges to the spectrograph’s configurations (especiallygrating positions). This is referred to as an “attached”ThAr. This is generally true for visitor-mode observationsbut, in service mode at the VLT, this is not the defaultoperation and grating positions are normally reinitializedbetween the quasar and ThAr calibration exposure; see thediscussion in Molaro et al. (2013).By comparing VLT observations of reflected sunlight from

asteroids with solar line atlases, Molaro et al. (2008) foundthat the ThAr wavelength scale was accurate enough forvarying-constant studies. A similar calibration techniqueusing quasars or stars observed through an iodine gas cellidentified substantial distortions of the wavelength scale onscales associated with echelle orders (Griest et al., 2010;Whitmore, Murphy, and Griest, 2010). However, these “intra-order” distortions were repeated from order to order, withoutany evidence of a long-range component. Therefore, H2

transitions would have effectively random shifts addedbetween them, but no significant systematic effect shouldbe seen in analyses using large numbers of transitions, like thevarying μ work (Murphy, Webb, and Flambaum, 2009;Whitmore, Murphy, and Griest, 2010).Unfortunately, more recent investigations using archival

VLT and Keck spectra of asteroids and stars with solarlikespectra (“solar twins”) found that substantial long-rangedistortions in ThAr wavelength scales are actually the norm(Rahmani et al., 2013; Whitmore and Murphy, 2015); theearlier works appear simply to have sampled unrepresentativeperiods when these distortions were very small. The effect ofthese long-range distortions on μ-variation measurementscan be understood from Fig. 6: the K coefficients of the

Lyman-band transitions decrease systematically toward longerwavelengths. Therefore, long-range distortions of the quasarwavelength scale will cause a shift in Δμ=μ. The recentinvestigations of μ variation with VLT and Keck spectra haveshown these long-range distortions to be the main systematicuncertainty in measuring Δμ=μ accurately. Several of theseanalyses have been able to use this “supercalibration” infor-mation to correct the distortions in the spectra, with varyingdegrees of confidence due to lack of knowledge about how thedistortions change on short time scales (or spectrograph andtelescope configurations), e.g., the studies of HE0027 − 1836(Rahmani et al., 2013), B0642 − 5038 (Bagdonaite, Ubachset al., 2014), J1237þ 064 (Daprà et al., 2015), and J1443þ2724 (Bagdonaite et al., 2015). Virtually all the monitoredlong-range wavelength distortions for the VLT caused apositive shift in the value of Δμ=μ by a few 10−6 (i.e.,correcting for the distortions gave a more negative Δμ=μ).

B. Fitting of spectra

The first method, referred to as a line-by-line or reducedredshift method, isolates narrow regions in the spectrumwhere single H2 absorption lines occur, without any or verylittle visible overlap from Lyman-α forest and metal lineabsorptions (Ivanchik et al., 2005; Reinhold et al., 2006;Thompson et al., 2009; Wendt and Molaro, 2011, 2012). Peakpositions are then fitted to determine an observed redshift zi ¼λzi =λ

0i − 1 for each transition which is then implemented in a

fitting routine, minimizing deviations to Eq. (4) and determin-ing values for Δμ=μ and zabs. After the fit the reduced redshift

ζi ¼zi − zabs1þ zabs

ð10Þ

can be calculated. An advantage of this method is that itprovides simple, visible insight into the effect of a varying μby plotting the reduced redshift ζi as a function of thesensitivity coefficient Ki for each line. In view of the relation

ζi ¼Δμμ

Ki; ð11Þ

the slope of the plot directly represents the Δμ=μ value, whileoutliers in the data points are easily detected. However, theline-by-line method is suited only for spectra containingisolated, singular absorption features of H2, as is the casefor system Q0347 − 383 (Wendt and Molaro, 2011, 2012).Even then, an absorption profile that appears singular might infact be a sum of multiple narrowly separated velocitycomponents, and the only robust means for testing thatpossibility is to fit all H2 transitions simultaneously, as inthe “comprehensive fitting” method discussed later and usedfor most analyses to date.Complex velocity profiles can be disentangled using a

different fitting method which is known as the comprehensivefitting method. Within this approach, all available transitionsare fitted simultaneously and solved for a single redshift foreach identifiable absorbing H2 velocity component; see, e.g.,King et al. (2008), Malec et al. (2010), and Bagdonaite,Ubachs et al. (2014). Spectral regions where H2 profiles are



partially overlapped by the Lyman-α forest, metal absorptionlines, or other H2 transitions can be handled in this way aswell, leading to an increased amount of information extractedfrom a single quasar spectrum. This is particularly the case ifthe H2 profile is broad as demonstrated in the analysis ofQ2348 − 011; see Fig. 8 and Bagdonaite et al. (2012).The software package2 VPFIT is commonly used to

implement the comprehensive fitting analysis. VPFIT is anonlinear least-squares χ2-minimization program dedicatedspecifically for analyzing quasar spectra composed ofcomplex overlaps of multiple Voigt profiles. The analysisstarts by selecting a set of spectral regions where the H2

transitions can be visually discerned from the Lyman-αforest. In each region, a different effective continuum is setby the broad forest lines but an H2 velocity structure is seenrepetitively and, therefore, fitting it requires fewer freeparameters compared to the case of unrelated lines. For asingle H2 velocity component, all the transitions share thesame redshift zabs, width b, and the same column density NJfor transitions from the same rotational level J. By invokingsuch a parameter linking, one is able to minimize the totalnumber of free parameters for H2 and to gain informationfrom multiple regions simultaneously. The laboratory wave-lengths λ0i , the oscillator strengths fi, and the dampingparameters Γi are the fixed parameters of the fit which areknown from laboratory measurements or molecular physicscalculations (see Sec. III).The comprehensive fitting method permits fitting complex

velocity structures. The necessity to add more componentsto an H2 profile is evaluated by inspecting the residuals offitted regions individually and combined. The latterapproach results in a “composite residual spectrum”(Malec et al., 2010) where normalized residuals of multipleregions are aligned in velocity or redshift space andaveraged together. Any consistent underfitting is seen moreclearly in the composite residual spectrum than in individualresiduals. Naturally, various goodness-of-fit indicators (e.g.,a reduced χ2) are also used to justify the addition of morecomponents. Once a satisfactory fit is achieved, a final freeparameter Δμ=μ is introduced which permits relative shiftsof the H2 transitions according to their Ki coefficients. Asingle Δμ=μ value is then extracted from the multiple fittedtransitions of H2.

C. Known H2 absorption systems at moderate-to-high redshift

In Table I a listing is given of moderate-to-high redshiftH2 absorption systems as obtained from the literature. Themajority of these quasars were observed with UVES-VLT,while system J2123 − 005 was observed at both Keck andVLT, J0812þ 3208 was observed at Keck, and the lowerredshift systems HE0515 − 441 and Q1331þ 170 wereobserved with the space telescope imaging spectrographaboard the Hubble Space Telescope. Some of the lowerredshift systems, Q1331þ 170 and Q1441þ 014, wereobserved at VLT specifically for the electronic COabsorptions. The table is restricted to H2 absorption

FIG. 8. The H2 transitions L0R0 indicated with solid (blue)markers and L0R1 indicated with dashed (green) markers seen atdifferent redshifts toward ten quasars. Transitions are imprinted ina number of velocity features as indicated. For further details seethe main text.2http://www.ast.cam.ac.uk/~rfc/vpfit.html



http://www.ast.cam.ac.uk/%7Erfc/vpfit.html






TABLE I. List of known moderate-to-high redshift H2 absorption systems with some relevant parameters. Bessel R magnitude taken from theSuperCOSMOS Sky Survey (Hambly et al., 2001). The ten systems analyzed so far for μ variation are offset at the top. The column densitiesNðH2Þ, NðHDÞ, NðCOÞ, and NðHIÞ are given on a log10 scale in cm−2. Some systems were investigated for CO only, and hence no columndensities determined for H and H2. It is noted that the 23 systems listed in Table I of Balashev et al. (2014) should also be regarded as very likelyH2 absorption systems.

Quasar zabs zem RA(J2000) Decl.(J2000) NðH2Þ NðHDÞ NðCOÞ NðHÞ Rmag Ref.

HE0027 − 1836 2.42 2.55 00∶30∶23.62 −18∶19∶56.0 17.3 21.7 17.37 Noterdaeme, Ledoux et al.(2007), Rahmani et al. (2013)

Q0347 − 383 3.02 3.21 03∶49∶43.64 −38∶10∶30.6 14.5 20.6 17.48 Ivanchik et al. (2005), Reinholdet al. (2006) King et al. (2008),Wendt and Molaro (2011,2012), Thompson et al. (2009)

Q0405 − 443 2.59 3.00 04∶07∶18.08 −44∶10∶13.9 18.2 20.9 17.34 Ivanchik et al. (2005), Reinholdet al. (2006), King et al. (2008),Thompson et al. (2009)

Q0528 − 250 2.81 2.81 05∶30∶07.95 −25∶03∶29.7 18.2 13.3 21.1 17.37 King et al. (2008, 2011)B0642 − 5038 2.66 3.09 06∶43∶26.99 −50∶41∶12.7 18.4 21.0 18.06 Noterdaeme, Ledoux et al.

(2008), Bagdonaite, Ubachset al. (2014), Albornoz Vásquezet al. (2014)

Q1232þ 082 2.34 2.57 12∶34∶37.58 þ07∶58∶43.6 19.7 15.5 20.9 18.40 Varshalovich et al. (2001),Ivanchik et al. (2010)

J1237þ 064 2.69 2.78 12∶37∶14.60 þ06∶47∶59.5 19.2 14.5 14.2 20.0 18.21 Noterdaeme et al. (2010)J1443þ 2724 4.22 4.42 14∶43∶31.18 þ27∶24∶36.4 18.3 21.0 18.81 Ledoux, Petitjean, and Srianand

(2006), Bagdonaite et al. (2015)J2123 − 0050 2.06 2.26 21∶23∶29.46 −00∶50∶52.9 17.6 13.8 19.2 15.83 Malec et al. (2010),

van Weerdenburg et al. (2011)Q2348 − 011 2.42 3.02 23∶50∶57.87 −00∶52∶09.9 18.4 20.5 18.31 Ledoux et al. (2006),

Noterdaeme, Petitjean et al.(2007), Bagdonaite et al. (2012)

J0000þ 0048 2.52 3.03 00∶00∶15.17 þ00∶48∶33.29 15 19.2 Noterdaeme (2015)Q0013 − 004 1.97 2.09 00∶16∶02.40 −00∶12∶25.0 18.9 20.8 17.89 Petitjean, Srianand, and Ledoux

(2002)Q0201þ 113 3.39 3.61 02∶03∶46.66 þ11∶34∶45.4 ≤16.4 21.3 19.41 Srianand et al. (2012)HE0515 − 441 1.15 1.71 05∶17∶07.63 −44∶10∶55.5 19.9 14.0 Reimers et al. (2003)Q0551 − 366 1.96 2.32 05∶52∶46.18 −36∶37∶27.5 17.4 20.5 17.79 Ledoux, Srianand, and Petitjean

(2002)J0812þ 3208 2.63 2.70 08∶12∶40.68 þ32∶08∶08.6 19.9 15.4 21.4 17.88 Tumlinson et al. (2010),

Balashev, Ivanchik, andVarshalovich (2010),Jorgenson, Wolfe, andProchaska (2010)

J0816þ 1446 3.29 3.85 08∶16∶34.39 þ14∶46∶12.5 18.66 22.0 19.20 Guimarães et al. (2012)Q0841þ 129 2.37 2.48 08∶44∶24.24 þ12∶45∶46.5 14.5 20.6 17.64 Petitjean, Srianand, and Ledoux

(2000)J0857þ 1855 1.72 1.89 08∶57∶26.79 þ18∶55∶24.4 13.5 17.32 Noterdaeme et al. (2011)J0918þ 1636 2.58 3.07 09∶18∶26.1 þ16∶36∶09 ≤19.0 21.0 19.49 Fynbo et al. (2011)J1047þ 2057 1.77 2.01 10∶47∶05.8 þ20∶57∶34 14.7 19.96 Noterdaeme et al. (2011)Q1331þ 170 1.78 2.08 13∶33∶35.81 þ16∶49∶03.7 19.7 14.8 21.2 16.26 Tumlinson et al. (2010),

Cui et al. (2005)J1337þ 3152 3.17 3.17 13∶37∶24.69 þ31∶52∶54.6 14.1 21.4 18.08 Srianand et al. (2010)J1439þ 1117 2.42 2.58 14∶39∶12.04 þ11∶17∶40.5 19.4 14.9 13.9 20.1 18.07 Noterdaeme, Petitjean et al.

(2008), Srianand et al. (2008)Q1444þ 014 2.08 2.21 14∶46∶53.04 þ01∶13∶56.0 18.3 20.1 18.10 Ledoux, Petitjean, and Srianand

(2003)J1456þ 1609 3.35 3.68 14∶56∶46.481 þ16∶09∶39.31 17.1 21.7 19.05 Noterdaeme et al. (2015)J1604þ 2203 1.64 1.98 16∶04∶57.49 þ22∶03∶00.7 14.6 19.09 Noterdaeme et al. (2009)J1705þ 3543 2.04 2.02a 17∶05∶42.91 þ35∶43∶40.3 14.1 19.42 Noterdaeme et al. (2011)Q2100 − 0641 3.09 3.14 21∶00∶25.029 −06∶41∶45.99 18.8 21.05 17.52 Balashev et al. (2015)J2140 − 0321 2.34 2.48 21∶40∶43.016 −03∶21∶39.29 20.1 22.4 18.93 Noterdaeme (2015), Balashev

et al. (2015)Q2318 − 111 1.99 2.56 23∶21∶28.69 −10∶51∶22.5 15.5 20.7 17.68 Noterdaeme, Ledoux et al.

(2007)J2340 − 0053 2.05 2.09 23∶40∶23.67 −00∶53∶27.00 18.2 20.35 17.66 Jorgenson, Wolfe, and

Prochaska (2010), Balashevet al. (2015)

Q2343þ 125 2.43 2.52 23∶46∶25.42 þ12∶47∶43.9 13.7 20.4 20.18 Petitjean et al. (2006),Dessauges-Zavadsky et al.(2004)

azem reported by Hewett and Wild (2010) is smaller than zabs from Noterdaeme et al. (2011).



systems with z > 1. Besides those detections, some tenlower redshift (0.05 ≲ z ≲ 0.7) molecular hydrogen absorb-ers have been discovered as well in the archival HubbleSpace Telescope and cosmic origins spectrographspectra [at logN½H2=cm−2� > 14.4 (Muzahid, Srianand,and Charlton, 2015)].Even though H2 is the most abundant molecule in the

Universe, its detections are rather scarce outside the localgroup of galaxies. As a rule, molecular hydrogen absorption isassociated with DLAs which are large reservoirs of neutralhydrogen gas with logN½HI=cm−2� ≥ 20.3. In a recent studyof 86 medium-to-high resolution quasar spectra that containDLAs, only a 1% (<6% at 95% confidence) detection rate ofH2 was reported for logN½H2=cm−2� > 17.5 (Jorgensonet al., 2014).The most recent quasar census from the Sloan Digital Sky

Survey contains over 80 000 spectra of high-redshift quasars(Pâris et al., 2014) that were searched for H2 absorption. Theintermediate spectral resolution of ∼2000, the usually lowsignal-to-noise ratio (SNR), and the presence of the Lyman-αforest overlapping the H2 region complicate the identificationof H2 absorption systems. However, Balashev et al. (2014)devised an automated search and identification procedure,including a quantitative assessment of the H2 detectionprobability as well as a false detection estimate. This pro-cedure yielded the identification of 23 highly likely H2

absorption systems at column densities logN½H2=cm−2� >18.8 in the redshift interval z ¼ 2.47–3.68. As an example, thesystem J2347 − 005 at z ¼ 2.59 displays a robust and unam-biguously assignable H2 absorption spectrum. For concisenessthis sample of 23 additional systems is not copied into Table I,but it should definitely be regarded as part of the pool ofpossible targets for future investigations of μ variation basedon the H2 method.The 33 systems listed in Table I added with the 23 very

likely additional systems identified by Balashev et al. (2014)make up a currently available database of 56 H2 knownabsorption systems that may be used for searches of μ varia-tion. However, not every H2 detection proves to be equallyuseful in the analysis of μ variation—some can be discardedbecause observational requirements are currently too chal-lenging or particular absorbing systems have unsuitableintrinsic properties. An ideal system should have thefollowing:

• A bright background source so that a high-resolutionspectrum with a signal-to-noise ratio in the continuum ofabout 50 per 2.5 km=s pixel could be obtained within,e.g., 15–20 h at an 8–10 m class optical telescope. Thiscan be achieved for background quasars with Rmag ¼17–18.

• A column density of H2 in the range betweenlogN½H2=cm−2� ∼ 14 and 18. A column density outsidethis range would yield either a small number of detect-able H2 transitions or a high number of saturatedtransitions; neither of these situations is desirable sincethe precision at which Δμ=μ can be measured dependson the number of strong but unsaturated transitions. Acolumn density as high as logN½H2=cm−2� ∼ 18–19,however, has the advantage that H2 transitions from

high-J states in combination with larger numbers of HDand CO lines would become observable at a high signal-to-noise ratio.

• An absorption redshift of at least z ¼ 2 or larger to assurethat a sufficient number of lines (typically > 40) willshift beyond the atmospheric cutoff at 3000 Å.

• Absorption profiles of H2 with simple substructure.While fitting multiple Voigt components to an H2

absorption profile is feasible and can be justified, asimpler absorption substructure is preferred because itsimplifies the analysis and tests for systematic errors.

A system that obeys these requirements yields a Δμ=μconstraint with a precision of several parts per million. Table Icontains a list of the ten best H2 systems that have alreadybeen analyzed for μ variation, and 23 additional systemswhose properties seem less suitable for a varying-μ analysis,although not all have been investigated in sufficient detail.Relevant properties of all systems, such as absorption andemission redshifts, position on the sky, the known columndensities for H2, deuterated molecular hydrogen HD, carbonmonoxide CO, and neutral atomic hydrogen H, and themagnitude are provided in the table. With similar Ki sensi-tivities as those of H2, the rovibronic transitions of HD(Ivanov et al., 2010) and CO (Salumbides et al., 2012)provide a way to independently cross check Δμ=μ constraintsfrom H2. However, the column densities of HD and CO areusually ∼105 times smaller than NðH2Þ, leading to a muchsmaller number of detections and fewer transitions in case ofdetection.

D. Constraints from individual quasars

A comparison between the ten quasar absorption spectra,for which the H2 absorption spectrum is analyzed in detail, isdisplayed in Fig. 8. In each case the wavelength region of twoH2 absorption lines is covered, the L0R0 line at a restwavelength of 1108.13 Å and the L0R1 line at 1108.63 Å,plotted on a wavelength scale and corresponding velocityscale (in km/s). The spectra illustrate typical characteristics ofhigh redshift H2 absorption. First of all this figure exemplifiesthe fact that the H2 spectra for different absorbers fall indifferent wavelength ranges. The L0R0 and L0R1 lines, at thered side of the H2 absorption spectrum, are detected at 3390 Åin the ultraviolet for the lowest redshift system J2123 − 005,to 5789 Å in the yellow range for the highest redshift systemin the sample. The velocity structure varies strongly from oneabsorber to the next, displaying single velocity features as inB0642 − 5038 and HE0027 − 1836, to three features as inQ0528 − 250 and J1443þ 2724, to an amount of sevenclearly distinguishable features in the case of Q2348 − 011.In more detailed analyses of the spectra additional underlyingvelocity components are revealed. The widths for both L0R0and L0R1 lines in Q1232þ 082 are broadened due tosaturation of the absorption in view of the large columndensity. Other lines in this sight line exhibit a width of4.5 km=s. The broad absorption lines due to the Lyman-αforest are random, with two strong HI lines appearing in thedisplayed wavelength interval toward B0642 − 5038, one ofthem overlapping the L0R1 line. Similarly in the spectrum ofJ1443þ 2724 all three velocity features of the L0R1 line are



hidden by a Lyman-α forest line. For Q0347 − 383 both L0R0and L0R1 lines are covered by broad forest features. Here it ispointed out that the number density of Lyman-α forest featuresincreases with increasing redshift (Meiksin, 2009) making theanalysis of absorption systems at higher z gradually moredifficult.Details on the ten H2 absorption systems for which a value

for Δμ=μ was deduced are as follows:• HE0027 − 1836: This system at z ¼ 2.40 was observedover three years at VLT and an analysis involvingcalibrations with attached ThAr spectra as well asasteroid spectra, providing a correction for long-rangewavelength distortions, yields a constraint of Δμ=μ ¼ð−7.6� 8.1stat � 6.3systÞ × 10−6 (Rahmani et al., 2013),which corresponds to Δμ=μ ¼ ð−7.6� 10.2Þ × 10−6.

• Q0347 − 383: First rounds of analysis for this system atz ¼ 3.02 were pursued by Ivanchik et al. (2002, 2005)and Reinhold et al. (2006) based on a line-by-line analysis method, in part relying on less accuratelaboratory reference data from classical spectroscopy.Subsequently the comprehensive fitting method wasemployed by King et al. (2008) yielding Δμ=μ ¼ð8.5� 7.4Þ × 10−6. Advanced line-by-line reanalysisstudies of the existing data for this object yieldedΔμ=μ ¼ð−28� 16Þ × 10−6 (Thompson et al., 2009), while re-newed observations yielded Δμ=μ ¼ ð15� 9stat�6systÞ × 10−6 (Wendt and Molaro, 2011) and Δμ=μ ¼ð4.3� 7.2Þ × 10−6 (Wendt and Molaro, 2012). Aweighted average over the latter fourmost accurate resultsyields Δμ=μ ¼ ð5.1� 4.5Þ × 10−6.

• Q0405 − 443: Similarly as for the previous item someearly studies were performed on this system at z ¼ 2.59(Ivanchik et al., 2005; Reinhold et al., 2006). A studyusing the comprehensive fitting method (King et al.,2008) yielded Δμ=μ ¼ ð10.1� 6.2Þ × 10−6 and a line-by-line fitting study (Thompson et al., 2009) yieldedΔμ=μ ¼ ð0.6� 10Þ × 10−6. As an average value weadopt Δμ=μ ¼ ð7.5� 5.3Þ × 10−6. Although this systemexhibits two absorption features (see Fig. 8), the weakerone was left out in all μ-variation analyses performedso far.

• Q0528 − 250: This system at z ¼ 2.81 was subject to thefirst constraint on a varying μ with observations from theMMT by Foltz, Chaffee, and Black (1988) yieldingΔμ=μ < 2 × 10−4. Further early stage observations wereperformed with the Keck telescope by Cowie andSongaila (1995). Based on observations at VLT (Ledoux,Petitjean, and Srianand, 2003) an accurate analysis wasperformed using the comprehensive fitting methodyielding Δμ=μ ¼ ð−1.4� 3.9Þ × 10−6 (King et al.,2008). Later renewed VLT observations were conductedwith ThAr attached calibrations yielding Δμ=μ ¼ ð0.3�3.2stat � 1.9systÞ × 10−6 (King et al., 2011). As anaverage value we adopt Δμ=μ ¼ ð−0.5� 2.7Þ × 10−6.

• B0642 − 5038: This system at z ¼ 2.66 at the mostsouthern declination (dec ¼ −50°) exhibits a single H2

absorption feature that was analyzed in a line-by-lineanalysis, yielding Δμ=μ¼ð7.4�4.3stat�5.1systÞ×10−6

(Albornoz Vásquez et al., 2014) and in a comprehensivefitting analysis, yielding Δμ=μ ¼ ð12.7� 4.5stat �4.2systÞ × 10−6 (Bagdonaite, Ubachs et al., 2014). Thelatter study includes a correction for long-range wave-length distortions by comparing to asteroid and solar-twin spectra. From these studies we adopt an averagedvalue Δμ=μ ¼ ð10.3� 4.6Þ × 10−6.

• Q1232þ 082: This system at z ¼ 2.34 exhibits strongabsorption in a single feature with strongly saturatedlines for low-J entries. From over 50 absorption lines aselection was made of 12 isolated, unsaturated, andunblended lines that were compared in a line-by-lineanalysis with two laboratory wavelength sets (nowoutdated) to yield Δμ=μ ¼ ð14.4� 11.4Þ × 10−5 andΔμ=μ ¼ ð13.2� 7.4Þ × 10−5 (Ivanchik et al., 2002),which averages to Δμ=μ ¼ ð140� 60Þ × 10−6. Thisconstraint is much less tight than from the other studiesin view of the limited data set of unsaturated lines. It istherefore not included in Fig. 9, although the result isincluded in the calculation of the total average.

• J1237þ 064: This system at z ¼ 2.69 exhibits threedistinct H2 absorption features, one of which is stronglysaturated for the low-J components. An analysis of over100 lines of H2 and HD yields a value of Δμ=μ ¼ð−5.4� 6.3stat � 3.5systÞ × 10−6 (Daprà et al., 2015).This value represents a result after invoking a long-range distortion-correction analysis. Combining the

FIG. 9. Values for Δμ=μ obtained for nine H2 absorptionsystems analyzed as a function of redshift and look-back timeplotted alongside with the evolution of cosmological parametersfor Ωm (dashed line) and ΩΛ (solid line), referred to the right-hand vertical axis. These are the dark-matter and dark energydensities, respectively, relative to the critical density. Open circlesrefer to data where a correction for long-range wavelengthdistortions is included; for the filled (red) circles such correctionhas not been applied. Uncertainties for individual points areindicated at the 1σ level by vertical line extensions to the points.The result for system Q1232þ 082 is not plotted in view of itslarge uncertainty. The gray bar represents the average for Δμ=μwith �1σ uncertainty limits from the 10 H2 systems analyzed.For comparison results from radio-astronomical observations atlower redshift (z < 1) are plotted as well (blue diamonds).



uncertainties we adopt the valueΔμ=μ ¼ ð−5.4� 7.2Þ ×10−6 in our analysis.

• J1443þ 2724: This system at the highest redshift(z ¼ 4.22) and the most northern (dec ¼ 27°) probedwas observed with UVES-VLT. For the μ-variationanalysis archival data from 2004 (Ledoux et al., 2006)were combined with data recorded in 2013 yielding aconstraint from an analysis also addressing the problemof long-range wavelength distortions in the ThArcalibrations Δμ=μ ¼ ð−9.5� 5.4stat � 5.3systÞ × 10−6

(Bagdonaite et al., 2015), which corresponds toΔμ=μ ¼ ð−9.5� 7.5Þ × 10−6.

• J2123 − 005: This system at z ¼ 2.05was observed withthe highest resolution ever for any H2 absorption systemwith theHIRES spectrometer on theKeck telescope usinga slit width of 0.3 in. delivering a resolving power of110 000. A comprehensive fitting analysis yielded a valueof Δμ=μ ¼ ð5.6� 5.5stat � 2.9systÞ × 10−6 (Malec et al.,2010). Independently a spectrum was observed, in visitormode, using the UVES spectrometer at the VLT, deliv-ering a value of Δμ=μ ¼ ð8.5� 3.6stat � 2.2systÞ × 10−6

(van Weerdenburg et al., 2011). Averaging over theseindependent results yields Δμ=μ ¼ ð7.6� 3.5Þ × 10−6.

• Q2348 − 011: This system at z ¼ 2.43 exhibits a recordcomplex velocity structure with seven distinct H2 ab-sorption features and some additional underlying sub-structure. Nevertheless, a comprehensive fitting could beapplied to this system yielding a constraint of Δμ=μ ¼ð−6.8� 27.8Þ × 10−6 (Bagdonaite et al., 2012).

E. Current high-redshift constraint on Δμ=μ

For the ten systems analyzed, results for Δμ=μ weredetermined by various researchers. Statistical and systematicuncertainties were combined here and the results from thevarious analyses were averaged to yield a single value ofΔμ=μwith an associated 1σ uncertainty for each system. Theseresults for the ten H2 quasar systems are plotted in Fig. 9(except for Q1232þ 082) along with the more precise resultsfrom radio-astronomical observations at redshifts z ¼ 0.68and 0.89. The final result is obtained by taking a weightedaverage over the outcome for all ten systems in the intervalz ¼ 2.0–4.2, yielding Δμ=μ ¼ ð3.1� 1.6Þ × 10−6. This resultshows a larger proton-electron mass ratio in the past at asignificance of 1.9σ. We interpret this result as a null resultbearing insufficient significance to call for new physics interms of varying constants.It is further noted that for some of the results from

individual quasars the long-range wavelength distortionsfound recently in ESO-VLT spectra were addressed, by eithercorrecting for the distortion and/or as an increase of thesystematic uncertainty. This was done for systems HE0027 −1836 (Rahmani et al., 2013), B0642 − 5038 (Bagdonaite,Ubachs et al., 2014), J1443þ 2724 (Bagdonaite et al., 2015),and J1237þ 064 (Daprà et al., 2015). For the other systemsno such corrections were carried out, e.g., for the reason thatno supercalibration spectra are available or that at the time ofanalysis this systematic effect was not yet known, and thevalues adopted in Fig. 9 are as published. So for some of the

latter systems there may still exist the underlying systematiceffect connected to long-range wavelength distortions of thecalibration scale. From the distortion analyses carried out itfollows that for most cases the distortion gives rise to anincrease ofΔμ=μ by a few times 10−6, which is commensuratewith the finding of a positive value of similar size. Areanalysis of possible wavelength distortions on some systemswould be called for, but unfortunately solar-twin or asteroidspectra were not always recorded in exactly the time periodsof the quasar observations. This open-ended calibration issueprompts the assignment of an increased systematic error to thecombined result we presented earlier. Awaiting furtherdetailed assessment, we conservatively estimate a null resultof jΔμ=μj < 5 × 10−6 (3σ) for redshifts in the rangez ¼ 2.0–4.2.In Fig. 9 an explicit connection is made between varying

constants over a range of redshifts and the evolution of matterdensity in the Universe, a connection also made in theoreticalmodels predicting the temporal dependence of the values ofthe coupling constants. In view of the coupling of theadditional dilaton fields to matter, constants should not driftin a dark-energy-dominated universe (Barrow, Sandvik, andMagueijo, 2002; Sandvik, Barrow, and Magueijo, 2002;Barrow and Magueijo, 2005). All H2 data are obtained forredshifts z > 2, or look-back times in excess of 10 × 109 yr,in the epoch of a matter-dominated universe with Ωm > 0.9.According to prevailing theoretical models, if any variation ofconstants should occur, it should fall in this early stage ofevolution. It must be concluded that the H2 data of Fig. 9either do not comply with theoretical predictions of varyingconstants or that the experimental findings are not sufficientlyconstraining to verify or falsify the predictions.

F. Spatial effect on μ

An analysis of over 300 measurements of ionized metallicgas systems in quasar absorption spectra revealed a possiblespatial variation of the fine-structure constant α. This spatialdistribution effect was represented in terms of a dipole, as afirst order approximation of a spatial effect. The statisticalanalysis of the comprehensive data set yields the dipole to fallalong the axis given by coordinates RA ¼ 17.5� 0.9 h anddec ¼ −58� 9° at a 4.2σ significance level (Webb et al.,2011; King et al., 2012). The direction of this dipole axis Θα

for the spatial effect on α is indicated in Fig. 10. RecentlyWhitmore and Murphy (2015) argued that the long-rangewavelength distortions on the ThAr calibrations could be heldresponsible for at least part of the reported α-dipole phe-nomenon. In this section we explore a possible spatial effecton μ to be deduced from the analyzed hydrogen absorbers.Figure 10 graphically displays the spatial distribution of all

56 currently known H2 absorption systems plotted in anangular coordinate system [RA, dec]. The open (red) datapoints represent the ten sight lines for which detailed H2

analyses were performed, while the possible H2 target systemsare displayed as blue data points, dark blue for the 23 quasarslisted in Table I, and light blue for the 23 systems of Balashevet al. (2014). Inspection of this sky-map figure illustrates theremarkable feature that the ten investigated H2 absorbingsystems all fall in a narrow band across the sky, perpendicular



to the α-dipole angle Θα. This feature of a biased data setdictates that a search for a spatial effect in a three-dimensionalspace is severely hampered.Nevertheless, attempts to uncover a possible spatial effect

were pursued based on the data for Δμ=μ. First an uncon-strained fit was performed to extract an optimized dipole axisfrom the nine most precisely measured systems, withQ1232þ 085 being too imprecise. This yields an axis atRA ¼ 4.9� 4.8 h and dec ¼ −66� 30° and an amplitude ofð9.5� 5.3Þ × 10−6. The optimum values span a planeperpendicular to the dipole axis Θα, but the resulting valuesfor the angles are ill defined. This outcome of the fittingprocedure, which is likely to optimize for a spread of values intheir plane of reference, may indeed be regarded as an artifactof the biased data set.Second an attempt was made to assess whether the Δμ=μ

measurements are consistent with the putative α-dipole axisdetermined by Webb et al. (2011) at Θα ¼ ½RA ¼ 17.5 h,dec ¼ −58°]. For this purpose the projection angles Θμ of theH2 absorbers were calculated with respect to the axis Θα andplotted in Fig. 11. This plot illustrates once more that the H2

absorbers all lie at values of cos½Θμ − Θα� ≈ 0, hence Θμ⊥Θα.Subsequently a fit was performed to the functional form

Δμ=μ ¼ A0 þ AΘ cos½Θμ − Θα�; ð12Þ

yielding a dipole amplitude of AΘ ¼ ð2.39� 0.68Þ × 10−5,representing an effect at the 3σ significance level. Fitting apossible linear offset returns A0 ¼ ð0.6� 1.4Þ × 10−6 whichmakes a monopole contribution insignificant. Note that thebiased data apparently exaggerate a trend along theperpendicular direction toΘα. If anything, these analyses showthe importance of investigating H2 absorbing systems that fall

outside the band of the ten analyzed H2 absorber systems inorder to uncover possible spatial effects of μ variation.

V. PERSPECTIVES

The results on Δμ=μ obtained from the analysis of H2

spectra from VLT and Keck observations, now yielding a 3σconstraint jΔμ=μj < 5 × 10−6, are not easy to improve with

FIG. 10. Sky map in equatorial coordinates (J2000) showing currently known quasar sight lines containing molecular absorbers atintermediate-to-high redshifts. The open (red) points correspond to the ten H2 absorbers that have been analyzed for a variation of μ,while the dark (blue) points mark the 23 H2 targets that might be used in future analyses as listed in Table I. The gray (light blue) pointscorrespond to the additional sample of 23 H2 absorption systems found through the Sloan Digital Sky Survey (Balashev et al., 2014).The stars (green) indicate the sight lines toward PKS1830 − 211 and B0218þ 357, where μ variation was measured from CH3OH(Bagdonaite, Daprà et al., 2013; Bagdonaite, Jansen et al., 2013) and NH3 (Henkel et al., 2009), or only from NH3 (Murphy et al., 2008;Kanekar, 2011), respectively. The þ sign represents the angle Θα of the α-dipole phenomenon (Webb et al., 2011). The gray lineindicates the galactic plane.

FIG. 11. Values ofΔμ=μ of the results for H2 absorbers obtainedso far plotted with their angular coordinates in terms of aprojection angle Θ onto the dipole axis Θα (RA ¼ 17.5 h anddec ¼ −58°) associated with the dipole found for Δα=α (Webbet al., 2011). Open circles represent values of Δμ=μ that werecorrected for possible long-range wavelength distortions, whilesolid (red) circles refer to uncorrected data points. The solid linerepresents the result from a fit to Eq. (12) with the slope equalingAΘ. See the main text for several caveats in interpreting thesignificance of this fit.



existing technologies. The best known H2 absorption systems,fulfilling the requirements for obtaining high-quality spectra,now numbering 10 (see Table I), have been analyzed in detail.There exist additional high-redshift systems with H2 absorp-tion (also listed in Table I), but those exhibit less favorableparameters, i.e., either too low or too high molecular columndensities, have too low redshifts for obtaining a full H2

spectrum from Earth-based telescopes, or the backgroundquasar appears faint. Nevertheless, additional observationswith in-depth analyses will enlarge the database and increasethe statistical significance of the constraint on Δμ=μ in theredshift range z ¼ 2.0–4.2. First of all the systematic long-range wavelength distortions must be addressed by (i) observ-ing solar-twin and asteroid spectra attached to quasar spectraand ThAr calibrations, and therewith correct for the distor-tions, and (ii) find and repair the causes of the distortions inthe instruments.Major improvements can be gained from technological

advances, reducing the statistical and systematic uncertainties,which are almost equally limiting the accuracy at the level offew 10−6. The development of the next generation telescopeswith larger collecting dishes, i.e., the ESO “Europeanextremely large telescope” (E-ELT) with a 39 m primarymirror, the giant Magellan telescope (GMT) with a 24.5 mdish, both planned for the southern hemisphere, and the “thirtymeter telescope” (TMT) planned for the northern hemisphere,will result in an up to 20-fold increase of signal-to-noise ratioon the spectrum for similar averaging times. Such a gain willyield an immediate improvement of the statistical uncertainty.If currently we measure Δμ=μ with typical uncertainties of∼ð5–7Þ × 10−6, at a 20-fold increase of SNR a precision levelof ð0.3–0.4Þ × 10−6 could be reached. This is comparable withthe most precise Δμ=μ constraints obtained from radiomeasurements of methanol and ammonia at lower redshifts.Advances in light collection and an increase of statistical

significance should be accompanied by improvements on thewavelength calibration, the major source of systematic uncer-tainties. Currently, wavelength calibration is performed bycomparison with reference spectra from ThAr lamps. Thisapplication is hindered by saturation and asymmetries ofcalibration lines, but a careful selection of lines warrants areliable calibration for the present needs and level of accuracy(Murphy, Tzanavaris et al., 2007). The use of frequency-comblasers for calibration of astronomical spectrographs, nowbeing explored and implemented (Murphy et al., 2007;Steinmetz et al., 2008; Wilken et al., 2012), will providean ultrastable, controlled, and dense set of reference lines,signifying a major improvement over ThAr calibration. Thistechnical advance will lead to more tightly constrainingbounds on Δμ=μ.The recently encountered long-range wavelength-distortion

problems are likely caused by differences in the beam pointingbetween light from the quasar sources and light from theThAr-emission lamps, rather than from the spectroscopicaccuracy of ThAr lines. Differences in pointing between lightfrom distant point sources (quasars) and nearby laser sourcesmay similarly give rise to offsets in calibration. For this reasonnovel spectrographs will be built for ultimate pressure andtemperature control and, most importantly, are designed to be

fiber fed rather than slit based. This holds for the proposedinstruments on the E-ELT and GMT but also for the echellespectrograph for rocky exoplanet and stable spectroscopicobservations (ESPRESSO) (Pepe et al., 2010), to be mountedand operated on the existing VLT in 2016. This fiber-fed, cross-dispersed, high-resolution (resolving powerλ=Δλ ¼ 120 000 and a very high-resolution mode atλ=Δλ ¼ 220 000) echelle spectrograph will receive light fromeither one or all four VLTs via a coudé train optical system.This system will be ideally suited for investigation of high-redshift H2 quasar spectra. However, an issue with the fiberfeed to ESPRESSO and with most future fiber-fed spectro-graphs mounted on large telescopes is the wavelength cover-age. Fiber transmission in the range 3000–3800 Å is low, andthe current design window for ESPRESSO covers only 3800–6860 Å. This implies that a redshift as high as z ¼ 3.1 isrequired to observe the entire H2 spectrum until the Lymancutoff, while at the red end spectral lines at 1140 Å should becaptured within the window below 6860 Å, i.e., this is forredshifts z < 5. These limitations on redshift are ratherrestrictive in view of potential targets as listed in Table I.Next, one may consider the optimal resolution to be desired

for the H2 absorption method while using a spectrograph. Forthe existing 8–10 m optical telescopes, the designed propertiesfor the spectrograph are entirelymatched to the degree towhichlight from point sources like quasars are blurred out byturbulence and refraction effects in our atmosphere. Theangular extent of this seeing is typically 0.8–1.0 arc s projectedon the sky. The entrance slits for spectrographs are matched tothis size so that they deliver an instrumental resolution ofFWHM ∼ 6 km s−1, or resolving power R ∼ 50 000. This issufficient to resolve a single, isolated velocity component in anabsorption system with total Doppler parameter of b ¼ ffiffiffi

2p

σ ≈0.60 FWHM ∼ 3–4 km s−1 (where FWHM≡ 2

ffiffiffiffiffiffiffiffiffiffiffi2 ln 2

pσ) from

kinetic and turbulent motions.For a discussion of desired resolution it is illustrative to

compare observations of J2123 − 005 at two very differentresolution settings. This quasar was observed with the HIRESspectrometer at Keck at a resolution of R ¼ 110 000 (Malecet al., 2010) and with the UVES spectrometer at VLT at R ¼53 000 (van Weerdenburg et al., 2011). The extremely highresolution at HIRES could be achieved due only to unrivaledseeing conditions of 0.3 arc s. The spectra, shown in Fig. 12,demonstrate that the factor of 2 better resolution in theHIRES-Keck spectrum does not effectively produce narrowerlinewidths. In the case of J2123 − 005 two very strong andnarrow velocity features build the spectrum. However, it isunlikely that each of those features is made up of just a singlevelocity component. In reality, there may be several compo-nents separated by less than ∼2 km s−1, which are blendedtogether. So, even if each of them only has a Doppler bparameter of ∼0.5 km s−1 or less, they combine to make a fluxprofile that, even at the very highest resolution, would mostlikely not be different from that obtained with R ∼ 50 000.While this may be the case for most H2 absorbers, it is possiblethat some may have isolated features which truly are made upof just a single, narrow component. Possibly, one of theabsorbers HE0027 − 1836, Q0405 − 443, or Q0347 − 383might be such a case, which could be verified by recording



higher resolution spectra of them. The ESPRESSO spectro-graph, and also the Potsdam Echelle Polarimetric andSpectroscopic Instrument (PEPSI) (Strassmeier et al., 2015)with a resolution of R ¼ 270 000 planned for the largebinocular telescope, might be used for testing this hypothesis.If a quasar system would be found exhibiting narrower lineprofiles their analysis would definitely yield a more con-straining value for Δμ=μ, provided that a good signal-to-noiseratio can be obtained and calibration issues can be dealt with.But in most cases, if not all, a resolving power larger than50 000 will not by itself lead to significantly improvedconstraints on Δμ=μ.

VI. OTHER APPROACHES TO PROBE μ VARIATION

The focus of this Colloquium is on the application of the H2

method to observations of quasar spectra with Earth-basedtelescopes to detect a possible variation of the proton-electronmass ratio on a cosmological time scale. There are a number ofmethods and applications that are closely connected to thisquest. We selected four of those that will be discussed.

A. Gamma-ray bursts

Besides quasars, the brief and very luminous gamma-rayburst (GRB) afterglows can also be used in searches forextragalactic H2 absorption. GRBs are thought to mark theviolent death of very massive stars leaving black holes orstrongly magnetized neutron stars. Contrary to QSO sightlines, the GRB DLA usually refers to the (star-forming)host galaxy. Only four detections of H2 absorption in GRBDLAs are known so far, including GRB 080607 at z ¼ 3.04(Prochaska et al., 2009), GRB 120815A at z ¼ 2.36 (Krühleret al., 2013), GRB 120327A at z ¼ 2.81 (D’Elia et al., 2014),and GRB 121024A at z ¼ 2.30 (Friis et al., 2015). While the

quality of the current GRB spectra containing H2 is notsufficient to obtain competitive Δμ=μ constraints, GRBs canin principle give access to more H2 detections at redshifts ashigh as z≳ 4. Another practical advantage of GRB afterglowover QSO spectra is that their power-law spectra are feature-less. QSOs become very rare at redshifts larger than 6.

B. Chameleon fields: Environmental dependences of μ

There exist theoretical scenarios predicting fundamentalconstants to depend on environmental conditions, such assurrounding matter density or gravitational fields. The fieldsthat cause these effects go under the name of chameleontheories (Khoury and Weltman, 2004), signifying a distinctorigin of varying constants from dilaton field theories(Bekenstein, 1982; Barrow, Sandvik, and Magueijo, 2002;Sandvik, Barrow, and Magueijo, 2002). The hydrogen absorp-tion method can similarly be applied to probe the spectrum ofH2 under conditions of strong gravity as experienced in thephotospheres of white dwarfs, i.e., the compact electron-degenerate remnant cores of evolved low-mass stars thatexhibit at their surface a gravitational potential some ∼104times stronger than that at the Earth’s surface. In a few casesH2 molecules are found under such conditions (Xu et al.,2013). Observations of white dwarf objects in our Galaxy arenot subject to cosmological redshift, although there are smalleffects of gravitational redshift observable in the spectra.Again Lyman and Werner band lines are the characteristic

features, but at the prevailing temperatures of T ∼11 000–14 000 K the specific lines contributing to the spec-trum are decisively different. It is mainly due to the B −Xð1; v00Þ and B − Xð0; v00Þ Lyman bands for populated vibra-tional states v00 ¼ 3 and 4 that contribute to the absorptionspectrum. Galactic observation of these bands implies that theabsorption wavelengths occur in the vacuum ultraviolet,specifically in the interval λ ¼ 1295–1450 Å. Hence, obser-vations from outside the Earth’s atmosphere are required, andthese were performed with the cosmic origins spectrographaboard the Hubble Space Telescope (Xu et al., 2013). Part of aspectrum of the white dwarf GD29-38 (WD2326þ 049) isshown in Fig. 13.A μ-variation analysis was performed for the white dwarf

spectra based on the same principles as for the quasarobservation (Bagdonaite, Salumbides et al., 2014). Thereare, however, remarkable differences mainly related to thedifference in excitation temperatures of the absorbing gas.While in quasar absorption systems some 100 lines areobserved in the case of the hot photospheres and over 1000lines contribute to the spectrum. A fingerprint spectrum isgenerated starting from a partition function for a certaintemperature T:

Pv;JðTÞ ¼gnð2J þ 1Þe−Ev;J=kTPvmax

v¼0

PJmaxðvÞJ¼0 gnð2J þ 1Þe−Ev;J=kT

; ð13Þ

where Ev;J are excitation temperatures of ðv; JÞ quantumlevels, k is the Boltzmann constant, and gn is a nuclear-spindegeneracy factor, for which we adopt a 3∶1 ratio for ortho:para levels. Each spectral line, populated by Pv;J, is then

FIG. 12. Comparison of a part of the spectrum of the absorptionsystem toward J2123 − 005 at z ¼ 2.05 by the Keck telescopeequipped with the HIRES spectrograph, indicated by a gray(green) line (Malec et al., 2010), recorded at a resolving power of110 000 and by the very large telescope, equipped with theUVES spectrograph indicated by a dark (black) line (vanWeerdenburg et al., 2011) at a resolving power of 53 000. Notethe absence of improved effective resolution in the Keckspectrum and the higher SNR in the UVES spectrum.



connected to a wavelength λi, an oscillator strength fi, a linedamping Γi, while for the entire ensemble a Doppler width bholds. The fingerprint spectrum is then included in a fit toEq. (4), introducing a sensitivity coefficient Ki for each line(Salumbides et al., 2015). The parameter z now represents thecombined effect of a Doppler velocity of the white dwarf starand a gravitational redshift. A simultaneous fit routine isdefined to optimize values of T, via Eq. (13), and of Δμ=μ andz via Eq. (4) along with a total H2 column density.For the two available white dwarf spectra, for GD29-38 and

GD133 obtained with the cosmic origins spectrograph (COS)aboard the HST (Xu et al., 2013), this procedure resulted inbounds on the μ variations ofΔμ=μðGD29–38Þ < ð−5.8� 3.7Þ ×10−5 and Δμ=μðGD133Þ < ð−2.3� 4.7Þ × 10−5 (Salumbideset al., 2015). The observations provide information on afundamental constant at a certain value for the gravitationalpotential, which may be expressed in dimensionless units as

ϕ ¼ GMRc2

ð14Þ

withG the Newton-Cavendish gravitational constant,M and Rthe mass and the radius of the white dwarf, and c the speed oflight. Note that in these dimensionless units the gravitationalpotential ϕ is equal to the gravitational redshift for lighttraveling out of the white dwarf photosphere z ¼ GM=Rc2.The prevailing conditions for the two white dwarfs at theirsurface are ϕðGD29–38Þ ¼ 1.9 × 10−4 and ϕðGD133Þ ¼1.2 × 10−4, where the condition at the Earth’s surface isϕEarth ¼ 0.69 × 10−9. The presently found result may beinterpreted as a bound on a dependence of the proton-electronmass ratio jΔμ=μj < 4 × 10−5 at field strengths of ϕ > 10−4.

When compared to constraints from Earth-bound experimentwith atomic clocks, taking advantage of the ellipticity of theorbit around the Sun, the H2 white dwarf studies are lessconstraining (Blatt et al., 2008; Tobar et al., 2013), but bearthe advantage of probing regions of much stronger gravity.

C. μ variation in combination with other fundamental constants

Besides μ and α, various combinations of dimensionlessconstants have been probed via spectroscopy of the extra-galactic interstellar and/or intergalactic medium. For example,comparing the ultraviolet (UV) transitions of heavy elementswith the hyperfine HI transition at a 21-cm rest wavelengthallows one to extract limits on x ¼ α2gp=μ, where gp is thegyromagnetic factor of the proton (Wolfe, Brown, andRoberts, 1976; Tzanavaris et al., 2005). To make such acomparison, one must take into account the fact that quasarscan have a frequency-dependent structure, which may result indifferent sight lines being probed at the UV, optical, and radiowavelengths. An application of this particular method led to ajΔx=xj constraint at the level of ≲1.7 × 10−6 for redshift z ¼3.174 (Srianand et al., 2010). The problem of a frequency-dependent source structure is mitigated by observation of theconjugate Λ-doublet transitions in the OH radical (Chengalurand Kanekar, 2003; Darling, 2003), or by the combination ofOH lines with the nearby HI 21-cm line to constraincombinations of fundamental constants μ, α, and gp(Kanekar et al., 2012). The latter study yields a constraintof jΔF=Fj ¼ ð−5.2� 4.3Þ × 10−6, with F ¼ gp½μα2�1.57 at anintermediate redshift of z ¼ 0.765.Comparison of nearby lying spectral lines of the CO

molecule (the 7-6 rotational transition) and the CI (2-1)fine-structure line, both observed in emission toward thelensed galaxy HLSJ091928.6þ 514223, constrain the pos-sible variation of another combination of dimensionlessconstants. Where the CO molecular line exhibits a linearsensitivity to the μ variation, the atomic carbon fine-structuresplitting exhibits a sensitivity to α2. The combined analysis ofthe two lines sets a bound on the parameter F ¼ α2=μ ofjΔF=Fj < 2 × 10−5 at a redshift of z ¼ 5.2 (Levshakov et al.,2012). Similar less-constraining studies on the CO/CI combi-nation have been performed as well, even up to redshift z ¼6.42 (Levshakov et al., 2008) and over the interval z ¼2.3–4.1 (Curran et al., 2011). Although this method bears theprospect of providing tight constraints on varying constants, itrelies on the assumption that CO and CI are cospatial.

D. Radio astronomy

Molecules exhibit many low-frequency modes associatedwith vibration, rotation, and inversion motion. While purerotational motion exhibits a sensitivity coefficient of K ¼ −1and vibrational motion of K ¼ −1=2 there are many examplesof sensitivity enhancement due to fortuitous degeneracies inthe quantum level structure of molecules (Kozlov andLevshakov, 2013; Jansen, Bethlem, and Ubachs, 2014).There are two prominent examples of importance to searchfor μ variation via radio astronomy: the ammonia (NH3) andthe methanol (CH3OH) molecules.

FIG. 13. Part of the absorption spectrum of the white dwarfG29-38 as obtained with the COS-HST in the range 1340–1370 Å. The solid (red) line is the result of a fit with optimizedparameters for T, b, N, z, and Δμ=μ. The tick (blue) marksrepresent the positions of H2 absorption lines contributing to thespectrum.



In NH3, the inversion tunneling of the N atom through theplane of the H atoms gives rise to transitions that scale with μas Einv ∼ μ−4.46 (van Veldhoven et al., 2004; Flambaum andKozlov, 2007). The sensitivity coefficient of K ¼ −4.46makes ammonia a 2 orders of magnitude more sensitiveprobe for varying μ than H2. A disadvantage of the ammoniamethod is that all inversion lines exhibit a similar sensitivity,hence a comparison with different molecules must be made inorder to solve for a degeneracy with redshift. Usually,the Ki ¼ −1 rotational transitions of other species, asHC3N and HCOþ, are picked as the reference transitions.The assumption that the multiple species occupy exactly thesame physical location and physical conditions is then asource of systematic uncertainty. Currently, two examplesof NH3 absorption are known outside the local universe: atz ¼ 0.69 toward B0218þ 357 and at z ¼ 0.89 towardPKS1830 − 211. Measurements of NH3 resulted in 3σ limitsof jΔμ=μj < 1.8 × 10−6 in the former sight line (Murphyet al., 2008) and <1.4 × 10−6 in the latter (Henkelet al., 2009).Methanol is an even more sensitive probe of varying μ, with

Ki sensitivities ranging from −33 to þ19 (Jansen et al., 2011;Levshakov, Kozlov, and Reimers, 2011). Besides purelyrotational transitions, the microwave spectrum of CH3OHincludes rotation-tunneling transitions arising via the hinderedrotation of the OH group, thus inducing the greatly enhancedsensitivity to μ. In fact, the most stringent current jΔμ=μjconstraints at the level of <1.1 × 10−7 (1σ) are derived from acombination of these CH3OH absorption lines at redshiftz ¼ 0.89 toward PKS1830 − 211 (Ellingsen et al., 2012;Bagdonaite, Daprà et al., 2013; Bagdonaite, Jansen et al.,2013; Kanekar et al., 2015). As opposed to the NH3 method,where a comparison with other molecular species is oftenused, methanol offers an advantage of a test based on a singlespecies. Unfortunately methanol has been detected only in thesingle lensed galaxy PKS1830 − 211, where it is subject to anumber of disturbing phenomena such as time variability ofthe background blazar and chromatic substructure (Muller andGuélin, 2008; Bagdonaite, Daprà et al., 2013).As of now only two lensed galaxies have been found with a

sufficient radiation level and column density to detect NH3

and CH3OH. Apart from PKS1830 − 211 and B0218þ 357,there exist three further intermediate redshift radio sourceswhere molecules have been detected, i.e., PKS1413þ 357

with absorption at z ¼ 0.247 (Wiklind and Combes, 1994),B1504þ 377 with absorption at z ¼ 0.672 (Wiklind andCombes, 1996), and PMN J0134 − 0931 with absorption atz ¼ 0.765 (Kanekar et al., 2005). PMN J0134 − 0931 is also agravitationally lensed object. The high detection rate ofgravitationally lensed objects is a selection bias becauselensing amplifies the flux of the background quasar. It isexpected that with the recently inaugurated and planned radio-astronomical facilities, such as the Atacama large millimeter/submillimeter array (ALMA, frequency range 30 GHz–1 THz) or the square kilometer array (SKA, frequency rangeup to 30 GHz in the final stage), more detections ofextragalactic molecular absorption will be found in theforeseeable future. The enhanced sensitivity of the new cm-and mm-wave telescopes will allow finding weaker radio

sources, and the broad bandwidths of the new spectrometerswill permit sensitive studies of varying constants at highredshift.

VII. CONCLUSION

The H2 quasar absorption method is demonstrated to be aviable method to search for putative variations of the proton-electron mass ratio μ, in particular, for redshift ranges z > 2.The method is based on a comparison between laboratory andastronomical absorption spectra of Lyman and Werner lines inmolecular hydrogen. The wavelengths of the laboratory linesare determined at the accuracy level of Δλ=λ < 10−7 and maytherefore be considered exact for the purpose of thesecomparisons, i.e., not contributing to the uncertainty onΔμ=μ. Sensitivity coefficients, representing the differentialwavelength offset of the spectral lines induced by a shift in μ,are calculated to an accuracy of<1% so that they do not causea limitation to a determination of Δμ=μ. By now ten H2

absorption systems have been analyzed in detail. They deliveran averaged quantitative result of Δμ=μ ¼ ð3.1� 1.6Þ × 10−6

in the redshift interval z ¼ 2.0–4.2. Part of this positive valueat the 1.9σ level might be explained by some not yet fullyincluded long-range wavelength distortions in the astronomi-cal calibrations. Hence we conservatively interpret this resultas a null result of jΔμ=μj < 5 × 10−6 (3σ), meaning that theproton-electron mass ratio has changed by less than 0.0005%for look-back times of ð10–12.4Þ × 109 yr into cosmic history.Attempts to interpret the results from the ten H2 absorbers

in terms of a spatial variation of μ along the dipole axis(RA ¼ 17.5 h and dec ¼ −58°), found from analyzing 300metal absorption systems for α at high redshift, did not yield areliable result because of the small sample available and thecoincidental alignment of these systems in a narrow bandacross the sky.Prospects of improving results by the H2 method are

identified. The upcoming opportunities of larger class tele-scopes (E-ELT, GMT, and TMT) will deliver increased photoncollection, where signal-to-noise improvements will straight-forwardly return tighter bounds on Δμ=μ. Most important is,however, improvement of calibration methods for the astro-nomical data. The combination of frequency-comb calibrationwith a fiber-fed spectrometer, as already planned for theESO-VLT (ESPRESSO spectrometer), would resolve theexisting calibration issues. Included coverage of the short-wavelength range [3000–3800] Å would be much desiredsince many of the identified high-quality H2 absorptionsystems (in the redshift range z ¼ 2–3) have their relevantfeatures in this window.As a final note we state that even incremental improvements

setting boundaries on drifting fundamental constants areworthwhile to pursue, given the importance of this endeavorinto the nature of physical law: Is it constant or not?

ACKNOWLEDGMENTS

The Netherlands Foundation for Fundamental Research ofMatter (FOM) is acknowledged for financial support throughits program “Broken Mirrors & Drifting Constants.” W. U.received funding from the European Research Council (ERC)



under the European Union’s Horizon 2020 research andinnovation program (Grant No. 670168). M. T. M. thanksthe Australian Research Council for Discovery Project GrantNo. DP110100866 which supported this work.

REFERENCES

Aaronson, M., J. H. Black, and C. F. McKee, 1974, Astrophys. J.191, L53.

Abgrall, H., E. Roueff, and I. Drira, 2000, Astron. Astrophys. Suppl.Ser. 141, 297.

Abgrall, H., E. Roueff, F. Launay, J. Y. Roncin, and J. L. Subtil,1993a, Astron. Astrophys. Suppl. Ser. 101, 273.

Abgrall, H., E. Roueff, F. Launay, J. Y. Roncin, and J. L. Subtil,1993b, Astron. Astrophys. Suppl. Ser. 101, 323.

Abgrall, H., E. Roueff, F. Launay, J. Y. Roncin, and J. L. Subtil,1993c, J. Mol. Spectrosc. 157, 512.

Albornoz Vásquez, D., H. Rahmani, P. Noterdaeme, P. Petitjean, R.Srianand, and C. Ledoux, 2014, Astron. Astrophys. 562, A88.

Bagdonaite, J., M. Daprà, P. Jansen, H. L. Bethlem, W. Ubachs, S.Muller, C. Henkel, and K. M. Menten, 2013, Phys. Rev. Lett. 111,231101.

Bagdonaite, J., P. Jansen, C. Henkel, H. L. Bethlem, K. M. Menten,and W. Ubachs, 2013, Science 339, 46.

Bagdonaite, J., M. T. Murphy, L. Kaper, and W. Ubachs, 2012, Mon.Not. R. Astron. Soc. 421, 419.

Bagdonaite, J., E. J. Salumbides, S. P. Preval, M. A. Barstow, J. D.Barrow, M. T. Murphy, and W. Ubachs, 2014, Phys. Rev. Lett. 113,123002.

Bagdonaite, J., W. Ubachs, M. T. Murphy, and J. B. Whitmore, 2014,Astrophys. J. 782, 10.

Bagdonaite, J., W. Ubachs, M. T. Murphy, and J. B. Whitmore, 2015,Phys. Rev. Lett. 114, 071301.

Bahcall, J. N., and M. Schmidt, 1967, Phys. Rev. Lett. 19, 1294.Bailly, D., E. Salumbides, M. Vervloet, and W. Ubachs, 2010, Mol.Phys. 108, 827.

Balashev, S. A., A. V. Ivanchik, and D. A. Varshalovich, 2010,Astron. Lett. 36, 761.

Balashev, S. A., V. V. Klimenko, A. V. Ivanchik, D. A. Varshalovich,P. Petitjean, and P. Noterdaeme, 2014, Mon. Not. R. Astron. Soc.440, 225.

Balashev, S. A., P. Noterdaeme, V. V. Klimenko, P. Petitjean, R.Srianand, C. Ledoux, A. V. Ivanchik, and D. A. Varshalovich,2015, Astron. Astrophys. 575, L8.

Barrow, J., H. Sandvik, and J. Magueijo, 2002, Phys. Rev. D 65,063504.

Barrow, J. D., 1987, Phys. Rev. D 35, 1805.Barrow, J. D., 2002, The Constants of Nature (Vintage Books,New York).

Barrow, J. D., and J. Magueijo, 2005, Phys. Rev. D 72, 043521.Barrow, J. D., H. B. Sandvik, and J. Magueijo, 2002, Phys. Rev. D65, 123501.

Bekenstein, J. D., 1982, Phys. Rev. D 25, 1527.Blatt, S., et al., 2008, Phys. Rev. Lett. 100, 140801.Born, M., 1935, Proc. Indian Acad. Sci. A 2, 533 [http://link.springer.com/article/10.1007%2FBF03045991].

Calmet, X., and H. Fritzsch, 2002, Eur. Phys. J. C 24, 639.Carlson, R. W., 1974, Astrophys. J. 190, L99.Carr, B. J., and M. J. Rees, 1979, Nature (London) 278, 605.Chengalur, J. N., and N. Kanekar, 2003, Phys. Rev. Lett. 91, 241302.Chupka, W. A., and J. Berkowitz, 1969, J. Chem. Phys. 51, 4244.Cowie, L. L., and A. Songaila, 1995, Astrophys. J. 453, 596.

Cui, J., J. Bechtold, J. Ge, and D. M. Meyer, 2005, Astrophys. J. 633,649.

Curran, S. J., A. Tanna, F. E. Koch, J. C. Berengut, J. K. Webb, A. A.Stark, and V. V. Flambaum, 2011, Astron. Astrophys. 533, A55.

Daprà, M., J. Bagdonaite, M. T. Murphy, andW. Ubachs, 2015, Mon.Not. R. Astron. Soc. 454, 489.

Darling, J., 2003, Phys. Rev. Lett. 91, 011301.Dekker, H., S. D’Odorico, A. Kaufer, B. Delabre, and H. Kotzlowski,2000, in Optical and IR Telescope Instrumentation and Detectors,edited by M. Iye and A. F. Moorwood, SPIE Conference SeriesVol. 4008 (SPIE–International Society for Optical Engineering,Bellingham, WA), pp. 534–545.

D’Elia, V., et al., 2014, Astron. Astrophys. 564, A38.Dent, T., and M. Fairbairn, 2003, Nucl. Phys. B653, 256.Dessauges-Zavadsky, M., F. Calura, J. X. Prochaska, S. D’Odorico,and F. Matteucci, 2004, Astron. Astrophys. 416, 79.

Dirac, P. A. M., 1937, Nature (London) 139, 323.Dzuba, V. A., V. V. Flambaum, and J. K. Webb, 1999, Phys. Rev.Lett. 82, 888.

Ellingsen, S. P., M. A. Voronkov, S. L. Breen, and J. E. J. Lovell,2012, Astrophys. J. Lett. 747, L7.

Flambaum, V. V., and M. G. Kozlov, 2007, Phys. Rev. Lett. 98,240801.

Flambaum, V. V., D. B. Leinweber, A.W. Thomas, and R. D. Young,2004, Phys. Rev. D 69, 115006.

Flambaum, V. V., and E. V. Shuryak, 2008, in Nuclei and MesoscopicPhysics, edited by P. Danielewicz, P. Piecuch, and V. Zelevinsky,AIP Conf. Proc. No. 995 (AIP, New York), pp. 1–11.

Flambaum, V. V., and A. F. Tedesco, 2006, Phys. Rev. C 73,055501.

Foltz, C. B., F. H. Chaffee, and J. H. Black, 1988, Astrophys. J. 324,267.

Friis, M., et al., 2015, Mon. Not. R. Astron. Soc. 451, 167.Fynbo, J. P. U., et al., 2011, Mon. Not. R. Astron. Soc. 413,2481.

Godun, R., P. Nisbet-Jones, J. Jones, S. King, L. Johnson, H.Margolis, K. Szymaniec, S. Lea, K. Bongs, and P. Gill, 2014,Phys. Rev. Lett. 113, 210801.

Griest, K., J. B. Whitmore, A. M.Wolfe, J. X. Prochaska, J. C. Howk,and G.W. Marcy, 2010, Astrophys. J. 708, 158.

Guimarães, R., P. Noterdaeme, P. Petitjean, C. Ledoux, R. Srianand,S. López, and H. Rahmani, 2012, Astron. J. 143, 147.

Hambly, N. C., et al., 2001, Mon. Not. R. Astron. Soc. 326, 1279.Hannemann, S., E. J. Salumbides, S. Witte, R. T. Zinkstok, E. J. vanDuijn, K. S. E. Eikema, and W. Ubachs, 2006, Phys. Rev. A 74,062514.

Henkel, C., K. M. Menten, M. T. Murphy, N. Jethava, V. V.Flambaum, J. A. Braatz, S. Muller, J. Ott, and R. Q. Mao, 2009,Astron. Astrophys. 500, 725.

Hewett, P. C., and V. Wild, 2010, Mon. Not. R. Astron. Soc. 405,2302.

Hollenstein, U., E. Reinhold, C. A. de Lange, and W. Ubachs, 2006,J. Phys. B 39, L195.

Huntemann, N., B. Lipphardt, C. Tamm, V. Gerginov, S. Weyers, andE. Peik, 2014, Phys. Rev. Lett. 113, 210802.

Ivanchik, A., P. Petitjean, D. Varshalovich, B. Aracil, R. Srianand, H.Chand, C. Ledoux, and P. Boissé, 2005, Astron. Astrophys. 440,45.

Ivanchik, A., E. Rodriguez, P. Petitjean, and D. A. Varshalovich,2002, Astron. Lett. 28, 423.

Ivanchik, A. V., P. Petitjean, S. A. Balashev, R. Srianand, D. A.Varshalovich, C. Ledoux, and P. Noterdaeme, 2010, Mon. Not. R.Astron. Soc. 404, 1583.



http://dx.doi.org/10.1086/181546

http://dx.doi.org/10.1086/181546

http://dx.doi.org/10.1051/aas:2000121

http://dx.doi.org/10.1051/aas:2000121

http://dx.doi.org/10.1006/jmsp.1993.1040

http://dx.doi.org/10.1051/0004-6361/201322544

http://dx.doi.org/10.1103/PhysRevLett.111.231101


http://dx.doi.org/10.1126/science.1224898

http://dx.doi.org/10.1111/j.1365-2966.2011.20319.x

http://dx.doi.org/10.1111/j.1365-2966.2011.20319.x



http://dx.doi.org/10.1088/0004-637X/782/1/10



http://dx.doi.org/10.1080/00268970903413350

http://dx.doi.org/10.1080/00268970903413350

http://dx.doi.org/10.1134/S1063773710110010

http://dx.doi.org/10.1093/mnras/stu275


http://dx.doi.org/10.1051/0004-6361/201425553

http://dx.doi.org/10.1103/PhysRevD.65.063504








http://link.springer.com/article/10.1007%2FBF03045991




http://dx.doi.org/10.1007/s10052-002-0976-0

http://dx.doi.org/10.1086/181515

http://dx.doi.org/10.1038/278605a0


http://dx.doi.org/10.1063/1.1671787

http://dx.doi.org/10.1086/176422

http://dx.doi.org/10.1086/444368

http://dx.doi.org/10.1086/444368

http://dx.doi.org/10.1051/0004-6361/201117457

http://dx.doi.org/10.1093/mnras/stv1998



http://dx.doi.org/10.1051/0004-6361/201323057

http://dx.doi.org/10.1016/S0550-3213(03)00043-9

http://dx.doi.org/10.1051/0004-6361:20034273

http://dx.doi.org/10.1038/139323a0



http://dx.doi.org/10.1088/2041-8205/747/1/L7




http://dx.doi.org/10.1103/PhysRevC.73.055501

http://dx.doi.org/10.1103/PhysRevC.73.055501

http://dx.doi.org/10.1086/165893

http://dx.doi.org/10.1086/165893


http://dx.doi.org/10.1111/j.1365-2966.2011.18318.x

http://dx.doi.org/10.1111/j.1365-2966.2011.18318.x


http://dx.doi.org/10.1088/0004-637X/708/1/158

http://dx.doi.org/10.1088/0004-6256/143/6/147

http://dx.doi.org/10.1111/j.1365-2966.2001.04660.x

http://dx.doi.org/10.1103/PhysRevA.74.062514


http://dx.doi.org/10.1051/0004-6361/200811475

http://dx.doi.org/10.1111/j.1365-2966.2010.16648.x

http://dx.doi.org/10.1111/j.1365-2966.2010.16648.x

http://dx.doi.org/10.1088/0953-4075/39/8/L02


http://dx.doi.org/10.1051/0004-6361:20052648

http://dx.doi.org/10.1051/0004-6361:20052648

http://dx.doi.org/10.1134/1.1491963

http://dx.doi.org/10.1111/j.1365-2966.2010.16382.x

http://dx.doi.org/10.1111/j.1365-2966.2010.16382.x

Ivanov, T. I., G. D. Dickenson, M. Roudjane, N. D. Oliveira, D.Joyeux, L. Nahon, W.-U. L. Tchang-Brillet, and W. Ubachs, 2010,Mol. Phys. 108, 771.

Ivanov, T. I., M. Roudjane, M. O. Vieitez, C. A. de Lange, W.-Ü. L.Tchang-Brillet, and W. Ubachs, 2008, Phys. Rev. Lett. 100,093007.

Ivanov, T. I., M. O. Vieitez, C. A. de Lange, and W. Ubachs, 2008,J. Phys. B 41, 035702.

Jansen, P., H. L. Bethlem, and W. Ubachs, 2014, J. Chem. Phys. 140,010901.

Jansen, P., L.-H. Xu, I. Kleiner, W. Ubachs, and H. L. Bethlem, 2011,Phys. Rev. Lett. 106, 100801.

Jennings, D. E., S. L. Bragg, and J. W. Brault, 1984, Astrophys. J.282, L85.

Jorgenson, R. A., M. T. Murphy, R. Thompson, and R. F. Carswell,2014, Mon. Not. R. Astron. Soc. 443, 2783.

Jorgenson, R. A., A. M. Wolfe, and J. X. Prochaska, 2010,Astrophys. J. 722, 460.

Kanekar, N., 2011, Astrophys. J. Lett. 728, L12.Kanekar, N., C. L. Carilli, G. I. Langston, G. Rocha, F. Combes, R.Subrahmanyan, J. T. Stocke, K. M. Menten, F. H. Briggs, and T.Wiklind, 2005, Phys. Rev. Lett. 95, 261301.

Kanekar, N., G. I. Langston, J. T. Stocke, C. L. Carilli, and K. M.Menten, 2012, Astrophys. J. Lett. 746, L16.

Kanekar, N., W. Ubachs, K. M. Menten, J. Bagdonaite, A.Brunthaler, C. Henkel, S. Muller, H. L. Bethlem, and M. Daprà,2015, Mon. Not. R. Astron. Soc. Lett. 448, L104.

Khoury, J., and A. Weltman, 2004, Phys. Rev. Lett. 93, 171104.King, J. A., M. T. Murphy, W. Ubachs, and J. K. Webb, 2011, Mon.Not. R. Astron. Soc. 417, 3010.

King, J. A., J. K. Webb, M. T. Murphy, and R. F. Carswell, 2008,Phys. Rev. Lett. 101, 251304.

King, J. A., J. K. Webb, M. T. Murphy, V. V. Flambaum, R. F.Carswell, M. B. Bainbridge, M. R. Wilczynska, and F. E. Koch,2012, Mon. Not. R. Astron. Soc. 422, 3370.

Klein, O., 1926, Z. Phys. 37, 895.Kolb, E. W., M. J. Perry, and T. P. Walker, 1986, Phys. Rev. D 33,869.

Komasa, J., K. Piszczatowski, G. Lach, M. Przybytek, B.Jeziorski, and K. Pachucki, 2011, J. Chem. Theory Comput. 7,3105.

Kozlov, M. G., and S. A. Levshakov, 2013, Ann. Phys. (Berlin) 525,452.

Krühler, T., et al., 2013, Astron. Astrophys. 557, A18.Landau, S. J., and G. Scóccola, 2010, Astron. Astrophys. 517, A62.Langacker, P., G. Segré, and M. J. Strassler, 2002, Phys. Lett. B 528,121.

Ledoux, C., P. Petitjean, J. P. U. Fynbo, P. Møller, and R. Srianand,2006, Astron. Astrophys. 457, 71.

Ledoux, C., P. Petitjean, and R. Srianand, 2003, Mon. Not. R. Astron.Soc. 346, 209.

Ledoux, C., P. Petitjean, and R. Srianand, 2006, Astrophys. J. 640,L25.

Ledoux, C., R. Srianand, and P. Petitjean, 2002, Astron. Astrophys.392, 781.

Levshakov, S. A., F. Combes, F. Boone, I. I. Agafonova, D. Reimers,and M. G. Kozlov, 2012, Astron. Astrophys. 540, L9.

Levshakov, S. A., M. G. Kozlov, and D. Reimers, 2011, Astrophys. J.738, 26.

Levshakov, S. A., D. Reimers, M. G. Kozlov, S. G. Porsev, and P.Molaro, 2008, Astron. Astrophys. 479, 719.

Lovis, C., and F. Pepe, 2007, Astron. Astrophys. 468, 1115.

Malec, A. L., R. Buning, M. T. Murphy, N. Milutinovic, S. L. Ellison,J. X. Prochaska, L. Kaper, J. Tumlinson, R. F. Carswell, and W.Ubachs, 2010, Mon. Not. R. Astron. Soc. 403, 1541.

Martins, C. J. A. P., E. Menegoni, S. Galli, G. Mangano, and A.Melchiorri, 2010, Phys. Rev. D 82, 023532.

Meiksin, A. A., 2009, Rev. Mod. Phys. 81, 1405.Meshkov, V. V., A. V. Stolyarov, A. V. Ivanchik, and D. A.Varshalovich, 2006, JETP Lett. 83, 303.

Mohr, P. J., B. N. Taylor, and D. B. Newell, 2012, Rev. Mod. Phys.84, 1527.

Molaro, P., S. A. Levshakov, S. Monai, M. Centurión, P. Bonifacio,S. D’Odorico, and L. Monaco, 2008, Astron. Astrophys. 481, 559.

Molaro, P., et al., 2013, Astron. Astrophys. 555, A68.Muller, S., and M. Guélin, 2008, Astron. Astrophys. 491, 739.Murphy, M. T., V. V. Flambaum, S. Muller, and C. Henkel, 2008,Science 320, 1611.

Murphy, M. T., P. Tzanavaris, J. K. Webb, and C. Lovis, 2007, Mon.Not. R. Astron. Soc. 378, 221.

Murphy, M. T., J. K. Webb, and V. V. Flambaum, 2003, Mon. Not. R.Astron. Soc. 345, 609.

Murphy, M. T., J. K. Webb, and V. V. Flambaum, 2009, Mem. Soc.Astron. Ital. 80, 833 [http://sait.oat.ts.astro.it/MSAIt800409/PDF/2009MmSAI..80..833M.pdf].

Murphy, M. T., J. K. Webb, V. V. Flambaum, J. X. Prochaska, andA. M. Wolfe, 2001, Mon. Not. R. Astron. Soc. 327, 1237.

Murphy, M. T., et al., 2007, Mon. Not. R. Astron. Soc. 380, 839.Muzahid, S., R. Srianand, and J. Charlton, 2015, Mon. Not. R.Astron. Soc. 448, 2840.

Noterdaeme, P., 2015 (private communication).Noterdaeme, P., C. Ledoux, P. Petitjean, F. Le Petit, R. Srianand, andA. Smette, 2007, Astron. Astrophys. 474, 393.

Noterdaeme, P., C. Ledoux, P. Petitjean, and R. Srianand, 2008,Astron. Astrophys. 481, 327.

Noterdaeme, P., C. Ledoux, R. Srianand, P. Petitjean, and S. Lopez,2009, Astron. Astrophys. 503, 765.

Noterdaeme, P., P. Petitjean, C. Ledoux, S. López, R. Srianand, andS. D. Vergani, 2010, Astron. Astrophys. 523, A80.

Noterdaeme, P., P. Petitjean, C. Ledoux, R. Srianand, and A.Ivanchik, 2008, Astron. Astrophys. 491, 397.

Noterdaeme, P., P. Petitjean, R. Srianand, C. Ledoux, and F. Le Petit,2007, Astron. Astrophys. 469, 425.

Noterdaeme, P., P. Petitjean, R. Srianand, C. Ledoux, and S. López,2011, Astron. Astrophys. 526, L7.

Noterdaeme, P., R. Srianand, H. Rahmani, P. Petitjean, I. Pâris, C.Ledoux, N. Gupta, and S. López, 2015, Astron. Astrophys. 577,A24.

Olive, K. A., M. Pospelov, Y.-Z. Qian, A. Coc, M. Cassé, and E.Vangioni-Flam, 2002, Phys. Rev. D 66, 045022.

Pachucki, K., 2010, Phys. Rev. A 82, 032509.Pâris, I., et al., 2014, Astron. Astrophys. 563, A54.Pepe, F. A., et al., 2010, Proc. SPIE Int. Soc. Opt. Eng. 7735, 77350F.Petitjean, P., C. Ledoux, P. Noterdaeme, and R. Srianand, 2006,Astron. Astrophys. 456, L9.

Petitjean, P., R. Srianand, and C. Ledoux, 2000, Astron. Astrophys.364, L26 [http://aa.springer.de/papers/0364001/2300l26/small.htm].

Petitjean, P., R. Srianand, and C. Ledoux, 2002, Mon. Not. R. Astron.Soc. 332, 383.

Philip, J., J. P. Sprengers, T. Pielage, C. A. de Lange, W. Ubachs, andE. Reinhold, 2004, Can. J. Chem. 82, 713.

Prochaska, J. X., et al., 2009, Astrophys. J. 691, L27.Rahmani, H., et al., 2013, Mon. Not. R. Astron. Soc. 435, 861.



http://dx.doi.org/10.1080/00268971003649307



http://dx.doi.org/10.1088/0953-4075/41/3/035702

http://dx.doi.org/10.1063/1.4853735

http://dx.doi.org/10.1063/1.4853735


http://dx.doi.org/10.1086/184311

http://dx.doi.org/10.1086/184311


http://dx.doi.org/10.1088/0004-637X/722/1/460

http://dx.doi.org/10.1088/2041-8205/728/1/L12


http://dx.doi.org/10.1088/2041-8205/746/2/L16

http://dx.doi.org/10.1093/mnrasl/slu206


http://dx.doi.org/10.1111/j.1365-2966.2011.19460.x

http://dx.doi.org/10.1111/j.1365-2966.2011.19460.x


http://dx.doi.org/10.1111/j.1365-2966.2012.20852.x

http://dx.doi.org/10.1007/BF01397481



http://dx.doi.org/10.1021/ct200438t

http://dx.doi.org/10.1021/ct200438t

http://dx.doi.org/10.1002/andp.201300010

http://dx.doi.org/10.1002/andp.201300010

http://dx.doi.org/10.1051/0004-6361/201321772

http://dx.doi.org/10.1051/0004-6361/201014215

http://dx.doi.org/10.1016/S0370-2693(02)01189-9

http://dx.doi.org/10.1016/S0370-2693(02)01189-9

http://dx.doi.org/10.1051/0004-6361:20054242

http://dx.doi.org/10.1046/j.1365-2966.2003.07082.x

http://dx.doi.org/10.1046/j.1365-2966.2003.07082.x

http://dx.doi.org/10.1086/503278

http://dx.doi.org/10.1086/503278

http://dx.doi.org/10.1051/0004-6361:20021187

http://dx.doi.org/10.1051/0004-6361:20021187

http://dx.doi.org/10.1051/0004-6361/201219042

http://dx.doi.org/10.1088/0004-637X/738/1/26

http://dx.doi.org/10.1088/0004-637X/738/1/26

http://dx.doi.org/10.1051/0004-6361:20079116

http://dx.doi.org/10.1051/0004-6361:20077249

http://dx.doi.org/10.1111/j.1365-2966.2009.16227.x



http://dx.doi.org/10.1134/S0021364006080017



http://dx.doi.org/10.1051/0004-6361:20078864

http://dx.doi.org/10.1051/0004-6361/201321351

http://dx.doi.org/10.1051/0004-6361:200810392


http://dx.doi.org/10.1111/j.1365-2966.2007.11768.x

http://dx.doi.org/10.1111/j.1365-2966.2007.11768.x

http://dx.doi.org/10.1046/j.1365-8711.2003.06970.x

http://dx.doi.org/10.1046/j.1365-8711.2003.06970.x

http://sait.oat.ts.astro.it/MSAIt800409/PDF/2009MmSAI..80..833M.pdf











http://dx.doi.org/10.1046/j.1365-8711.2001.04842.x

http://dx.doi.org/10.1111/j.1365-2966.2007.12147.x



http://dx.doi.org/10.1051/0004-6361:20078021

http://dx.doi.org/10.1051/0004-6361:20078780

http://dx.doi.org/10.1051/0004-6361/200912330

http://dx.doi.org/10.1051/0004-6361/201015147

http://dx.doi.org/10.1051/0004-6361:200810414

http://dx.doi.org/10.1051/0004-6361:20066897

http://dx.doi.org/10.1051/0004-6361/201016140

http://dx.doi.org/10.1051/0004-6361/201425376

http://dx.doi.org/10.1051/0004-6361/201425376



http://dx.doi.org/10.1051/0004-6361/201322691

http://dx.doi.org/10.1117/12.857122

http://dx.doi.org/10.1051/0004-6361:20065769

http://aa.springer.de/papers/0364001/2300l26/small.htm




http://dx.doi.org/10.1046/j.1365-8711.2002.05331.x

http://dx.doi.org/10.1046/j.1365-8711.2002.05331.x

http://dx.doi.org/10.1139/v04-042

http://dx.doi.org/10.1088/0004-637X/691/1/L27

http://dx.doi.org/10.1093/mnras/stt1356

Reimers, D., R. Baade, R. Quast, and S. A. Levshakov, 2003, Astron.Astrophys. 410, 785.

Reinhold, E., R. Buning, U. Hollenstein, A. Ivanchik, P. Petitjean,and W. Ubachs, 2006, Phys. Rev. Lett. 96, 151101.

Rosenband, T., et al., 2008, Science 319, 1808.Salumbides, E. J., J. Bagdonaite, H. Abgrall, E. Roueff, and W.Ubachs, 2015, Mon. Not. R. Astron. Soc. 450, 1237.

Salumbides, E. J., D. Bailly, A. Khramov, A. L. Wolf, K. S. E.Eikema, M. Vervloet, and W. Ubachs, 2008, Phys. Rev. Lett.101, 223001.

Salumbides, E. J., M. L. Niu, J. Bagdonaite, N. de Oliveira, D.Joyeux, L. Nahon, and W. Ubachs, 2012, Phys. Rev. A 86, 022510.

Sandvik, H. B., J. D. Barrow, and J. Magueijo, 2002, Phys. Rev. Lett.88, 031302.

Savedoff, M. P., 1956, Nature (London) 178, 688.Schellekens, A. N., 2013, Rev. Mod. Phys. 85, 1491.Shelkovnikov, A., R. J. Butcher, C. Chardonnet, and A. Amy-Klein,2008, Phys. Rev. Lett. 100, 150801.

Shlyakhter, A. I., 1976, Nature (London) 264, 340.Srianand, R., N. Gupta, P. Petitjean, P. Noterdaeme, and C. Ledoux,2010, Mon. Not. R. Astron. Soc. 405, 1888.

Srianand, R., N. Gupta, P. Petitjean, P. Noterdaeme, C. Ledoux, C. J.Salter, and D. J. Saikia, 2012, Mon. Not. R. Astron. Soc. 421, 651.

Srianand, R., P. Noterdaeme, C. Ledoux, and P. Petitjean, 2008,Astron. Astrophys. 482, L39.

Stadnik, Y. V., and V. V. Flambaum, 2015, Phys. Rev. Lett. 115,201301.

Staszewska, G., and L. Wolniewicz, 2002, J. Mol. Spectrosc. 212,208.

Steinmetz, T., et al., 2008, Science 321, 1335.Strassmeier, K. G., et al., 2015, Astron. Nachr. 336, 324.Sturm, S., F. Köhler, J. Zatorski, A. Wagner, Z. Harman, G. Werth,W. Quint, C. Keitel, and K. Blaum, 2014, Nature (London) 506,467.

Thompson, R. I., 1975, Astrophys. Lett. 16, 3.Thompson, R. I., J. Bechtold, J. H. Black, D. Eisenstein, X. Fan,R. C. Kennicutt, C. Martins, J. X. Prochaska, and Y. L. Shirley,2009, Astrophys. J. 703, 1648.

Thompson, R. I., J. Bechtold, J. H. Black, and C. Martins, 2009, NewAstron. 14, 379.

Tobar, M. E., et al., 2013, Phys. Rev. D 87, 122004.Tumlinson, J., et al., 2010, Astrophys. J. Lett. 718, L156.Tzanavaris, P., J. K. Webb, M. T. Murphy, V. V. Flambaum, and S. J.Curran, 2005, Phys. Rev. Lett. 95, 041301.

Ubachs, W., R. Buning, K. S. E. Eikema, and E. Reinhold, 2007,J. Mol. Spectrosc. 241, 155.

Ubachs, W., and E. Reinhold, 2004, Phys. Rev. Lett. 92, 101302.Uzan, J.-P., 2011, Living Rev. Relativity 14, 2.van Veldhoven, J., J. Küpper, H. L. Bethlem, B. Sartakov,A. J. A. van Roij, and G. Meijer, 2004, Eur. Phys. J. D 31,337.

van Weerdenburg, F., M. T. Murphy, A. L. Malec, L. Kaper, and W.Ubachs, 2011, Phys. Rev. Lett. 106, 180802.

Varshalovich, D. A., A. V. Ivanchik, P. Petitjean, R. Srianand, and C.Ledoux, 2001, Astron. Lett. 27, 683.

Varshalovich, D. A., and S. A. Levshakov, 1993, JETP Lett. 58,237.

Velchev, I., R. van Dierendonck, W. Hogervorst, and W. Ubachs,1998, J. Mol. Spectrosc. 187, 21.

Vogt, S. S., et al., 1994, in Instrumentation in Astronomy VIII, editedby D. L. Crawford and E. R. Craine, SPIE Conf. Ser. Vol. 2198(SPIE–International Society for Optical Engineering, Bellingham,WA), pp. 362–375.

Webb, J. K., V. V. Flambaum, C.W. Churchill, M. J. Drinkwater, andJ. D. Barrow, 1999, Phys. Rev. Lett. 82, 884.

Webb, J. K., J. A. King, M. T. Murphy, V. V. Flambaum, R. F.Carswell, and M. B. Bainbridge, 2011, Phys. Rev. Lett. 107,191101.

Wendt, M., and P. Molaro, 2011, Astron. Astrophys. 526, A96.Wendt, M., and P. Molaro, 2012, Astron. Astrophys. 541, A69.Whitmore, J. B., and M. T. Murphy, 2015, Mon. Not. R. Astron. Soc.447, 446.

Whitmore, J. B., M. T. Murphy, and K. Griest, 2010, Astrophys. J.723, 89.

Wiklind, T., and F. Combes, 1994, Astron. Astrophys. 286, L9.Wiklind, T., and F. Combes, 1996, Astron. Astrophys. 315, 86.Wilken, T., et al., 2012, Nature (London) 485, 611.Wolfe, A. M., R. L. Brown, and M. S. Roberts, 1976, Phys. Rev. Lett.37, 179.

Wolfe, A. M., E. Gawiser, and J. X. Prochaska, 2005, Annu. Rev.Astron. Astrophys. 43, 861.

Wolniewicz, L., T. Orlikowski, and G. Staszewska, 2006, J. Mol.Spectrosc. 238, 118.

Wolniewicz, L., and G. Staszewska, 2003, J. Mol. Spectrosc. 220, 45.Xu, S., M. Jura, D. Koester, B. Klein, and B. Zuckerman, 2013,Astrophys. J. Lett. 766, L18.

Xu, S. C., R. van Dierendonck, W. Hogervorst, andW. Ubachs, 2000,J. Mol. Spectrosc. 201, 256.



http://dx.doi.org/10.1051/0004-6361:20031313

http://dx.doi.org/10.1051/0004-6361:20031313









http://dx.doi.org/10.1038/178688b0



http://dx.doi.org/10.1038/264340a0

http://dx.doi.org/10.1111/j.1365-2966.2010.16574.x

http://dx.doi.org/10.1111/j.1365-2966.2011.20342.x

http://dx.doi.org/10.1051/0004-6361:200809727






http://dx.doi.org/10.1002/asna.201512172

http://dx.doi.org/10.1038/nature13026


http://dx.doi.org/10.1088/0004-637X/703/2/1648

http://dx.doi.org/10.1016/j.newast.2008.11.001

http://dx.doi.org/10.1016/j.newast.2008.11.001


http://dx.doi.org/10.1088/2041-8205/718/2/L156


http://dx.doi.org/10.1016/j.jms.2006.12.004


http://dx.doi.org/10.12942/lrr-2011-2

http://dx.doi.org/10.1140/epjd/e2004-00160-9

http://dx.doi.org/10.1140/epjd/e2004-00160-9


http://dx.doi.org/10.1134/1.1415856





http://dx.doi.org/10.1051/0004-6361/201014835

http://dx.doi.org/10.1051/0004-6361/201218862



http://dx.doi.org/10.1088/0004-637X/723/1/89

http://dx.doi.org/10.1088/0004-637X/723/1/89




http://dx.doi.org/10.1146/annurev.astro.42.053102.133950

http://dx.doi.org/10.1146/annurev.astro.42.053102.133950



http://dx.doi.org/10.1016/S0022-2852(03)00121-8

http://dx.doi.org/10.1088/2041-8205/766/2/L18


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